Quantum versus Chaos : Questions Emerging from Mesoscopic Cosmos (Fundamental Theories of Physics)

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Quantum versus Chaos

Fundamental Theories of Physics An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application

Editor:

ALWYN VAN DER MERWE University of Denver, U.S.A.

Editorial Advisory Board: LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University of Cambridge, U.K. CLIVE KILMISTER, University of London, U.K. PEKKA J. LAHTI, University of Turku, Finland GÜNTER LUDWIG, Philipps- Universität, Marburg, Germany ASHER PERES, Israel Institute of Technology, Israel NATHAN ROSEN, Israel Institute of Technology, Israel EDUARD PRUGOVECKI, University of Toronto, Canada MENDEL SACHS, State University of New York at Buffalo, U.S.A. ABDUS SALAM, International Centre for Theoretical Physics, Trieste, Italy HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der Wissenschafen, Germany

Volume 87

Quantum versus Chaos Questions Emerging from Mesoscopic Cosmos by

Katsuhiro Nakamura Faculty of Engineering, Osaka City University, Osaka, Japan

KLUWER ACADEMIC PUBLISHERS NEW YORK / BOSTON / DORDRECHT / LONDON / MOSCOW

eBook ISBN: Print ISBN:

0-306-47121-3 0-792-34557-6

©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

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Table of Contents Preface

ix

Chapter 1. Genesis of chaos and breakdown of quantization of adiabatic invariants 1.1. Introduction 1.2. Collapse of KAM tori and onset of chaos 1.3. Diagnostic characters of chaos 1.4. Suppression of chaos in quantum dynamics 1.5. Breakdown of quantization of adiabatic invariants References

1 1 2 7 9 11 14

Chapter 2. Semiclassical quantization of chaos: trace formula 2.1. Green's function and Feynman's path integral method 2.2. Quantization of integrable systems 2.3. Quantization of chaos: trace formula 2.4. Application of trace formula to autocorrelation functions 2.5. Significance and limitation of trace formula References

15 15 17 19 22 29 30

Chapter 3. Pseudo-chaos without classical counterpart in 1-dimensional quantum transport 3.1. Introduction 3.2. Quantum transport in superlattice and pseudo-chaos 3.3. Resonant tunneling in double-barrier structure and pseudo-chaos 3.4. General remarks References Chapter 4. Chaos and quantum transport in open magnetic billiards: from stadium to Sinai billiards 4.1. Introduction V

31 32 32 39 43 44

45 46

vi

Table of Contents 4.2. Magneto-conductance in stadium billiard: experimental results 4.3. Transition from chaos to tori 4.4. Quantum-mechanical and semiclassical theories 4.5. Comparison in stadium billiards between theory and experiment 4.6. Open Sinai billiard in magnetic field: distribution of Lyapunov exponents and ghost orbits 4.7. Comparison in Sinai billiard between quantal and classical theories 4.8. Summary References

47 50 54 59 66 70 75 76

Chapter 5. Chaotic scattering on hyperbolic billiards: success of semiclassical theory 5.1. Introduction 5.2. Exact quantum theory 5.3. Semiclassical theory 5.4. Distribution of complex resonances 5.5. Experiment on antidot arrays in magnetic field References

79 79 83 85 92 97 100

Chapter 6. Nonadiabaticity-induced quantum chaos 6.1. Avoided level crossings and gauge structure 6.2. Nonadiabatic transitions and gauge structure 6.3. Forces induced by Born-Oppenheimer approximation 6.4. Nonadiabaticity-induced chaos References

102 102 107 117 119 125

Chapter 7. Level dynamics and statistical mechanics 7.1. Level dynamics: from Brownian motion to generalized Calogero-Moser-Sutherland (gCM/gCS) system 7.2. Soliton turbulence: a new interpretation of irregular spectra 7.3. Statistical mechanics of gCM system 7.4. Statistical mechanics of gCS system in intermediate regime 7.5. Extension to case of several parameters References

127 127 136 142 151 154 160

Table of Contents Chapter 8. Towards time-discrete quantum mechanics 8.1. Stable and unstable manifolds in time-discrete classical dynamics 8.2. Breakdown of perturbation theory 8.3. Internal equation and Stokes phenomenon 8.4. Asymptotic expansion beyond all orders and homoclinic structures 8.5. Time discretization and quantum dynamics 8.6. Time-discrete unitary quantum dynamics 8.7. Time-discrete non-unitary quantum dynamics 8.8. Problems to be examined References

vii 162 162 166 168 175 181 183 186 196 197

Chapter 9. Conclusions and prospects References

198 207

Index

209

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Preface The framework of quantum mechanics in the adiabatic limit where no quantum transition occurs is traced back to the quantization condition of adiabatic invariants, i.e., of action variables. In fact, the interpretation of this condition as the commutation rule for a pair of canonical variables led to the construction of Heisenberg's matrix equation; another interpretation of this condition, as that for confining a standing wave, led to the birth of Schrödinger's wave mechanics. In classically-chaotic systems, however, the stable tori are broken up and we can conceive no action to be quantized. Therefore, we cannot prevent ourselves from being suspicious of using the present formalism of quantum mechanics beyond the logically acceptable (i.e., classicallyintegrable) regime. Actually, quantum dynamics of classically-chaotic systems yields only quasi-periodic and recurrent behaviors, thereby losing the classical-quantum correspondence. There prevails a general belief in the incompatibility between quantum and chaos. Presumably, a generalized variant of quantum mechanics should be established so as to accommodate the temporal chaos. The range of validity of the present formalism of quantum mechanics will be elucidated by an accumulation of experiments on the mesoscopic or nanoscale cosmos. Owing to recent progress in advanced technology, nanoscale quantum dots such as chaotic stadium and integrable circle billiards have been fabricated at interfaces of semiconductor heterojunctions, and quantum transport in these systems is under active experimental investigation. Anomalous fluctuation properties as well as interesting fine spectral structures that have already been reported are indicating symptoms of chaos. Quantum transport in mesoscopic systems will serve as a nice candidate for elucidating the effectivenes and noneffectiveness of quantum mechanics when applied to classically-chaotic systems. The experimental results could even provide a clue towards the creation of a generalized quantum mechanics, just as blackbody cavity radiation at the turn of the last century did for the creation of present-day ix

x

Preface

quantum mechanics. Therefore, in this book I shall investigate quantum transport in mesoscopic systems that are classically chaotic, showing the success and failure of theoretical trials to explain experimental issues. My basic idea is as follows: Our inability to explain anomalous quantum effects in mesoscopic systems is due partly to our formalism's inability to describe situations sensitive to initial conditions and partly to technological weakness in making fine-grained predictions without being affected by extrinsic noises and random potentials. Despite active research on the semiclassical quantum theory of chaotic systems, most of the semiclassical treatment of bounded and open systems have not fully succeeded to capture the clear signatures of chaos because of wave diffraction effects, the difficulty of systematic enumeration of scattering and/or periodic orbits, etc. I shall also develop the semiclassical theory (i.e., scattering theory for open systems and trace formula for bounded systems) and raise some unsatisfactory points involved in this traditional theory. The existing semiclassical theory could not be the ultimate theory of quantization of chaos. There is thus a need to go in a radically new direction to accommodate a genuine temporal chaos in quantum dynamics. In an attempt to see the unambiguous quantum-classical correspondence in the semiclassical realm of chaotic systems, we shall come to question the continuity of the time variable. With the help of recent progress in nonlinear classical dynamics, I have dared to hint at a slightly portentous proposal to construct a generalized quantum mechanics by discretizing the "time" and to describe interesting outcomes emerging from the procedure of time discretization in quantum dynamics. It is our hope that, through the insights gained from studying the chapters that follow, readers would be greatly encouraged to comprehend the incompatibility between quantum and chaos and to start their own speculation on a new framework of quantum mechanics that would unify these two key concepts in contemporary science. I am grateful to many people, including J. P. Bird, A. Bulgac, P. Gaspard, S. Kawabata, C. M. Marcus, S. A. Rice, and Y. Takane for stimulating discussions that have sharpened my ideas as embodied in the present book. I wish to thank Alwyn van der Merwe for his critical reading of the manuscript and improving its grammatical errata.

Chapter 1 Genesis of Chaos and Breakdown of Quantization of Adiabatic Invariants Key words and key concepts required to understand the following chapters are explained below. With increase of perturbations, resonances break the Kolmogorov-Arnold-Moser (KAM) tori, leading to a genesis of chaos. Characterization of chaotic behaviors is achieved by using Lyapunov exponent and the Kolmogorov-Sinai entropy. We consider how chaos affects quantum mechanics by addressing the breakdown in the quantization of adiabatic invariants.

1.1. Introduction Over the past decades, an increasing number of researchers have taken up studies of chaos. Most nonlinear dynamical systems, from driven pendulum to fluid turbulence, display chaotic behaviors. It is rather difficult to address, among diverse systems in nature, those that cannot exhibit chaos. The concept of chaos is, however, inherently relevant to classical dynamics. Standard diagnostic characters such as sensitivity to initial conditions and a nonvanishing Kolmogorov-Sinai entropy are meaningful only in classical dynamical systems. On the other hand, we all recognize that quantum mechanics, the greatest theory constructed in the 20th century, can explain a lot of microscopic phenomena, such as superconductivity, superfluidity, and the quantum Hall effect, and moreover serve as an indispensable 1

2

Chapter 1

guiding principle for today's science and technology. The genesis of chaos, however, is disturbing the foundation of the quantum theory. Researchers in the forefront have begun to reveal the quantummechanical fingerprints of chaos and even to contemplate the invention of a generalized version of quantum mechanics which would have an unambiguous correspondence with chaos. In classical mechanics, the Hamilton equation is nonlinear in general. In the case of chaotic systems, the stretching and folding (Smale's horse-shoe) mechanism gives rise to a phase droplet (i.e., a cluster of initial points in phase space) that evolves into self-similar structures on infinitely small scales in phase space. In the corresponding quantum dynamics, however, the wavefunctin will show a recurrent (time-periodic) phenomenon, i.e., suppression of chaotic diffusion because of the linearity of the time-dependent Schrödinger equation. From a viewpoint of measurement, Heisenberg's uncertainty principle imposes a limitation of the order of Planck constant in the resolution of phase space, leading to the unavoidable incompatibility between quanta and chaos. In this book we shall describe this incompatibility in detail and present some challenging attempt to reconcile or unify these contradictory concepts. To begin with, standard diagnostics of chaos will be sketched.

1.2. Collapse of KAM Tori and Onset of Chaos To explain the mechanism for the onset of chaos, we choose a twodimensional oscillator (without dissipation) described by the Hamiltonian (1.1) with the action Ji= A canonical transformation from {pi,qi} to and angle converts (1.1) into (1.2) =0. The {Ji} are obviously constants of motion, i.e., Any orbit is either periodic or quasi-periodic, and confined on the

Genesis of Chaos

3

surface of the torus characterized by a suitable set of radii J1 and J2; see Fig. 1.1. On introduction of nonintegrable perturbation V, the Hamiltonian becomes (1.3) where is a constant of O(1) and m, n run over the set of integers. The stability of the torus will be examined below. Using a generating function (1.4) we consider the canonical transformation (1.5a) (1.5b) Then the Hamiltonian (1.3) is transformed into

Fig. 1.1. 2-dimensional torus with action variables J1 and J2. mutually irreducible closed paths.

and

are

4

Chapter 1

(1.6) are exploited. The method for treating the term where = linear in depends on the magnitude of fmn If the magnitude of the perturbation is small enough to ensure > m + even for the nonresonant tori. Consequently one fails to get a generating function with . Then the probability density function for a system with N degrees of freedom is given by (1.12) which is a quantum analog of the classical distribution function in phase space.

10

Chapter 1

The problem of representation is important. By a Fourier transformation of the position representation of the density operator with respect to the relative coordinate one may obtain the Wigner representation of the wave function at p and q(=(q"+q'>/2) as

(1.13) While Pw(p,q) has a monumental significance, it can take negative values and show violent undulation of O( ) in phase space. Even in the semiclassical limit, therefore, the Wigner function in (1.13) can neither assimilate the classical distribution function (except for a very few linear systems like harmonic oscillators and noninteracting free particles) nor satisfy the Liouville equation even approximately. Because of its occasional negative values, Pw(p,q) does not qualify as a probability. This deficiency can be overcome by means of appropriate coarse graining guided by Heisenberg's uncertainty principle. Making a Gaussian smoothing of Pw(p,q) in (1.13) at every point (p,q) in phase space, we can finally arrive at (1.12). To make this statement concrete, a Gaussian wave packet will be chosen as an initial state. In general, up to the time of O( ), P(p,q) in (1.12) proves to mimic a (coarse-grained) classical distribution function, obeying the Liouville equation. In classically integrable and regular systems, the wave packet shows a simple (homogeneous or inhomogeneous) diffusion. In classically nonintegrable and chaotic systems, however, the profile of P(p,q) develops Smale's horse-shoe (i.e., stretching and folding) mechanism. Consequently, the wavepacket deforms to finer and finer textures, suggesting a formation of a fractal object. To proceed to a more quantitative description, we define contour lines C(t) and a phase space area enclosed by C such that the integrated probability takes a fixed (arbitrary) value. The area A (t) constitutes an incompressible phase liquid, in which every point executes its own classical motion. Corresponding to the wavepacket dynamics of the classically chaotic system, the pattern of A(t) deforms from a single spherical droplet to a finer and finer maze-like structure. The phase volume for the overall structure deduced by coarsening of fine textures is given by

(1.14)

Genesis of Chaos

11

This formula is a result of the fact that one direction of the phase liquid is maximally extended as t) due to the exponential growth in the difference of nearby orbits. On the other hand, Liouville’s theorem (i.e., the incompressibility of the phase liquid) imposes another direction orthogonal to to be contracted as exp( In quantum mechanics, however, there exists a lower limit in the resolution of phase space because of the uncertainty principle: The Therefore the linear dimension of each phase-space cell is of O classical-quantum correspondence is broken at the cross-over time, (1.15) when quantum dynamics inevitably fails to assimilate the classical dynamics any further. For quantum dynamics will develop interference between nearby fine textures with a resultant diffusion behavior thoroughly different from that for The argument above is justified for namely, so long as the similarity between P(p,q) in (1.12) and the (coarse-grained) classical distribution function is ensured up to the cross-over time. To conclude, the long-time quantum dynamics is governed by a quantum analog of Poincare' 's recurrence theorem: Both wavefunctions and energies reassemble themselves infinitely often in the course of long-time evolution. This phenomenon is called quantum recurrence.

1.5. Breakdown of Quantization of Adiabatic Invariants The onset of chaos will greatly affect quantum mechanics, which describes both bounded and open (scattering) systems. The BohrSommerfeld quantization condition for action lays the foundation of the present formalism of quantum mechanics in the limit where quantum transitions can be ignored. In fact, this condition, taken as the noncommutativity of canonical variables ) , led to the birth of Heisenberg’s matrix mechanics; the same condition, taken as that for the existence of a standing wave, following the de Broglie’s wave-particle dualism, gave rise to Schrödinger’s wave mechanics. The emergence of chaos, however, renders meaningless the quantization of action. The quantization condition for action is traced back to the

12

Chapter 1

experimental discovery of the quantization of adiabatic invariants. So, let us review a historical route to this discovery. Following Ehrenfest (1916) , we shall concentrate upon the problem of the radiation from a blackbody cavity. By the latter, we mean the cavity enclosed by a wall at temperature T that contains electromagnetic waves (i.e., an assembly of energy resonators) and emits radiation through a small hole to the outside. If one could move the wall adiabatically (very slowly) to expand or contract, the cavity volume V, frequency vi and energy E1 of each energy resonator would change as well. Einstein proved in 1911, however, that the ratio E /v

(1.16)

remains unchanged under the adiabatic change; this ratio is therefore called as the adiabatic invariant. On the other hand, another kind of adiabatic invariant found by Wien is v/T, which represents the displacement law. Combining these two invariants, one has the adiabatically-invariant equality E/v = F(v/T) for a given arbitrary function F(x). Noting the state density for the electromagnetic wave, the blackbody radiation rate in the frequency range is seen to obey a scaling formula: (1.17) Equation (1.17) was in fact verified by experiments. It should be emphasized that (1.17) was derived within a framework of classical theory. While Planck assumed E to be an integer multiple of hv to explain the experimental curve F in terms of statistical mechanics, this assumption implies the quantization of the adiabatic invariant in (1.16), i.e., E/v =nh with n =1,2,•••. The adiabatic invariant is thus a cornerstone leading to the birth of quantum theory. The quantization of the adiabatic invariant formally reduces to that of the action J= , since the adiabatic invariant turns out to be the action J (more precisely, 2 J). To state this explicitly, for a harmonic oscillator, with energy E=(p2+ w2q2) /2, we see that 2 J= pdq= area of ellipse =E /v Extending the quantization condition for action to systems with N (>1) degrees of freedom, one gets

Genesis of Chaos

13 (1.18)

with k =1,2•••,M and nk =0,1,2,•••. and mk represent mutuallyindependent closed paths (see Fig. 1.1) and Maslov index, respectively. In the completely-integrable case with the number of constants of motion M equal to N, N-dimensional tori are formed and all N actions {Jk} are calculable. We are then able to proceed to quantum theory. In nonintegrable case with M>1) is N(T) exp(hKST) /T (see Sinai, 1976), i.e., exponentially proliferating, while the amplitude of terms with a period T(= T ) is A (T) T T exp(-hKST/2). Hence the contribution of orbits with periods less than T is given by A(T) x N(T) exp(hKST/2), which brings about a serious problem of nonconvergence in (2.17). It is inevitable to devise a method to make (2.17) "conditionally convergent." To resolve this problem, new developments appeal to the theory of Riemann's zeta function together with the invention of a novel resummation of series expansion called a Riemann-Siege1 type resurgence (Berry and Keating, 1990). This idea can be applied to Gutzwiller's trace formula. By simple integration and exponentiation of (2.17), together with an expansion

in (2.19), one obtains (Gutzwiller, 1990) (2.20) where

(E) is the Ruelle zeta function, defined as (2.2 la)

with quantum weights (2.21b) From (2.21), one recognizes that poles of (E) lead to the quantum eigenvalue. Expanding the product sum in (2.21a) up to terms of relatively short periodic orbits, one obtains well-converging results. In fact, using periodic orbits thus organized, Gutzwiller applied his trace

22

Chapter 2

formula to the calculation of irregular spectra of donors with anisotropic effective mass in silicon, obtaining eigenvalues in good agreement with exact quantum eigenvalues. But the eigenvalues derived from poles of (E) are always accompanied by small imaginary components. Another formidable problem around the trace formula is to find all periodic orbits without missing any one of them. For some fully chaotic systems without any bifurcation, the symbolic coding of periodic orbits ensures the successful counting of all periodic orbits. For generic Hamiltonian systems, from which both chaos and KAM tori emerge, however, complexity in symbolic coding of periodic orbits prevents us from our reaching the formulae (2.20) and (2.21). Furthermore, some far more important issues should be addressed: The trace formula is valid up to the second leading order in No resurgence for suitable zeta functions can therefore produce eigenvalues with precisions higher than the order of N. If one were to resolve rigorously the problem of conditional convergence, the trace formula should be improved so as to incorporate all orders in . Such attempts, however, will lead us into the forest of complicated mathematics, which is not compatible with our aim to achieve simplicity of the fundamental law.

2.4. Application of Trace Formula to Autocorrelation Functions While the computation of the trace formula is practically difficult, the calculation of its autocorrelation function is feasible on having recourse to some approximate methods. These correlation functions are important in the physics of mesoscopic phenomena in the ballistic regime, where the elastic mean-free path of electrons is larger than the linear dimension of the system, and a deterministic law is operative. Below we shall apply the trace formula to persistent currents and S matrices in chaotic systems. Persistent Currents The correlation function of the persistent current was studied by several groups (Szafer and Altshuler, 1993; Berry and Keating, 1994). We shall here introduce Berry and Keating's work (1994). Owing to the Aharonov-Bohm effect, the persistent current appears in normal conductors in mesoscopic scales. Consider the noninteracting electron

Semiclassical Quantization of Chaos

23

gas confined in a conducting ring threaded by an Aharonov-Bohm magnetic flux . We choose the ring shape that guarantees complete chaos in the classical motion of a point particle. For instance, one may mention the Sinai billiard, i.e., a square conductor with a circular hollow in its inside (see Fig. 2.2). The contribution of a single-particle energy level En( ) to the persistent current is given by p dEn/d , where the mean density state ρ is the inverse of mean level separation. The flux will now be scaled by flux quanta =hc/e . Taking {En) as a set of energy levels near the Fermi level, the autocorrelation function of the persistent current in the ground state is given by (2.22) where ' integration and < >n imply the averages for one period and for levels lying below the Fermi level respectively.

Fig. 2.2. Aharonov-Bohm flux

and Sinai billiard with persistent current.

24

Chapter 2

Here we introduce the integrated density of states in terms of stair-case functions:

(2.23) In (2.23), the suffix signifies the additional procedure of coarsegraining the fine structures less than the mean level spacing ( =(2 p)-1) ; is just the density of states with its average given by ρ . With use of the identity the calculation of C( ) turns out to be reduced to that of the correlation function of

(2.24) where the average on E is that over levels below We proceed to apply the trace formula (2.17). Deriving the state density from (2.17) and integrating it over E, we find

j

(2.25) with Bj= (2 )-1 exp(iµ j){det( Mj - 1)}-½ ; {j } denotes periodic orbits. In (2.25) the action is supplemented by the Aharonov-Bohm term proportional to the winding number wj around the flux. The dependent factor, originating from the coarse-graining procedure, will render (2.25) absolutely convergent. On substitution of (2.25) into (2.24), one obtains double summations over periodic orbits. Once φ '-integration has been been carried out, however, diagonal terms alone survive: J

(2.26) If the summation in (2.26) is taken in order of increasing periods Tj , be replaced by

l l.

j

Assume a set of winding numbers {wj} for orbits with period T~T+dT to obey a Gaussian distribution with zero mean and with


25

variance of <wj2>= T/T0 , where T0 is the period of the shortest orbit. We can then carry out the T integration in (2.26), after replacing the wj -dependent factor in each of intervals T~T+dT by its Gaussian average. As a result we find (2.27) in the limit of the characteristic winding number >>1. For a ring in d dimensions, and ,yielding -(d-1) - (d-1) . Hence (2.27) shows asymptotic behaviors C(0)=w 2 and -1, respectively. C( in the limits of =O and > is guaranteed. In averaging G over (Fermi) energy, therefore, the last exponential factor in (2.36) can be regarded as a complex random number su with mean < su>=0 and variance < su su>= ss uu' . Accordingly, G has a vanishing mean value, and its variance is given by

(2.37) Replacing the sum in (2.37) over paths s by an integral over the (n, m-independent) area distribution proper to mixed systems as

(2.38a) with

(2.38b) one finds


29

Thus, for 2, conductance vs. magnetic field shows a feature of the same stochastic process as a fractional Brownian motion (Mandelbrot, 1982) with mean zero and and variance (see Fig. 2.3). Although examples in this section are intriguing, the semiclassical theory of chaotic scattering will not be satisfactory at all unless it can explain the transport properties in real experiments. Consider, for instance, the ballistic crossroad problem at the interface of GaAs/ AlGaAs heterostructures. We shall meet there the serious problem of wave diffraction at junction points between lead wires and the confining cavity region. The wave diffraction will affect locations of scattering resonances and thereby autocorrelation functions. Several other interesting experiments are also being designed that measure conductance and its fluctuations in mesoscopic solid state devices and aim at elucidating a quantum-mechanical symptom of chaos. In particular, as we shall describe in the later chapter, Marcus et al. (1992) exploited circle and stadium billiards at the interface of the above heterostructures, measuring the electric resistance as a function of magnetic field. The fluctuation features reported by them have a rich structure, which awaits challenging analyses beyond both the periodic orbit theory and the calculation of correlation functions: Both semiclassical and exact quantum-mechanical theories are not successful in explaining the complicated experimental results.

2.5. Significance and Limitation of Trace Formula

So long as one stays within the framework of Schrödinger-Feynman's quantum mechanics, Gutzwiller's trace formula remains one of the most valuable procedures for exposing the ambiguities obscuring the borderline between quantum and classical mechanics for chaotic systems. As seen in the previous section, the autocorrelation function of the trace formula can be calculated approximately. The calculation of the trace formula itself, however, will encounter serious fundamental problems. In generic systems with elliptic islands coexisting with chaotic sea, symbolic coding of periodic orbits is much less obvious.

30

Chapter 2

Even if one could achieve this coding, one would meet another difficulty, namely that of nonconvergence in the Gutzwiller series due to the exponential proliferation of periodic orbits. This problem may be partly overcome either by smoothing the density of states or by inventing a way of conditional convergence by means of the Ruelle zeta function. There remains, however, a more troublesome problem: For bounded systems, the eigenvalues computed from the trace formula are not real! To resolve this problem, one should improve the trace formula so as to include higher-order terms in which will demand more and more complicated mathematics. Since the fundamental law should be as simple as possible, we are here tempted to re-examine the validity of the Shrödinger-Feynman formalism of quantum mechanics for classically chaotic systems the hope of inventing its more general version.

References Berry, M. V., and Keating, J. P. (1990). J. Phys. A23, 4839. Berry, M. V., and Keating, J. P. (1994). J. Phys. A27, 6167. Blumel, R., and Smilansky, U. (1988). Phys. Rev. Lett. 60, 177. Gutzwiller, M. C. (1971). J. Math. Phys. 12, 343. Gutzwiller, M. C. (1990). Chaos in Classical and Quantum Mechanics. Berlin: Springer. Jalabert, R. A., Baranger, H. U., and Stone, A. D. (1990). Phys. Rev. Lett. 65, 2442. Ketzmerick, R. (1996). Preprint. Lai, Y. C., Blumel, R., Otto, E., and Grebogi, C. (1992). Phys. Rev. Lett. 68, 3491. Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. San Francisco: Freeman. Marcus, C. M., Rimberg, A. J., Westervelt, R. M., Hopkins, P. F., and Gossard, A. C. (1992). Phys. Rev. Lett. 69, 506. Miller, W. H. (1975). Adv. Chem. Phys. 30, 77. Sinai, Y. G. (1976). Introduction to Ergodic Theory. Princeton: Princeton University Press. Szafer, A., and Altshuler, B. L. (1993). Phys. Rev. Lett. 70, 587.

Chapter 3 Pseudo-Chaos Without Classical Counterpart in One-Dimensional Quantum Transport While there exists no genuine chaos in quantum systems due to the linearity of Schrödinger equation, we show in this chapter two interesting examples of pseudo-chaos in 1-dimensional (1-d) quantum transport. These phenomena arise from quantum-mechanical tunneling through potential barriers and have no classical counterpart. First we consider quantum transport in the 1-d Kronig-Penny model in a static electric field. The S matrix as a function of the number of barriers is examined in the complex plane. Although no positive Lyapunov exponent is available, it shows a stagnant chaos around a torus in the weak field case, while, in the strong field case, wandering from one stagnant region to another in an unpredictable way. The power spectra of transmission coefficients show a universal 1/f 2 behavior, which will be pointed out to be caused by a breakdown of the law of large numbers. Second, we shall investigate pseudo-chaos in multiple resonant tunnelings through a double-barrier structure. Both phenomena will be realized by using GaAs/AlAs heterostructures. The origin of pseudo- chaos will be revealed.

31

32

Chapter 3

3.1. Introduction Due to the linearity of the Schrödinger equation, the quantum system exhibits no chaos characterized by the standard diagnostics of Kolmogorov-Sinai entropy or positive Lyapunov exponents. We can therefore envisage merely a quantum analogue of chaos, i.e., quantum chaology, rather than genuine quantum chaos. Before beginning to elucidate many features of the quantum analog of chaos, we shall introduce interesting examples of pseudo-chaos in quantum systems which have no classical counterpart. Rapid progress in the fabrication of nanoscale structures has made it possible to see typical quantum-mechanical effects such as tunneling (Esaki, 1985; Mendez et al., 1988). In particular, growing attention has been paid to (1) superlattices (e.g., GaAs/AlAs) with alternating sequence of potential barriers and wells and (2) double-barrier structures through which resonant tunneling occurs. In this chapter, we shall first analyze pseudo-chaos accompanying quantum transport in a strictly regular superlattice, i.e., the KrönigPenny model, in a constant electric field (Nakamura et al., 1994) and then another pseudo-chaos in multiple resonant tunnelings through double-barrier structures (Jona-Lasinio et al., 1992).

3.2. Quantum Transport in Superlattice and Pseudo-Chaos Model Hamiltonian The superlattice is a nano-scale periodic structure composed of alternate arrays of potential barriers and quantum wells. For the 1-d superlattice with period and barrier width d in the presence of electric field (see Fig. 3.1), the Hamiltonian is given by (3.1) ( j-1) α +d and V(x)=0 with the periodic potential: V(x)=V0 for (j-1)α for (j-1)α+d with j=1,2, In (3.1), the last term of r.h.s. represents the coupling with an electric field For simplicity, is n

here replaced by a stair-wise function

(x- Ja), with θ (x)

= j- 1

denoting the step function.

Contrary

to some

recent studies

on

Pseudo-Chaos

33

Fig. 3.1. Kronig-Penny model in the presence of electric field. Inset: Tn in case of F=0. (k0=10 is used throughout in Figs. 3.1-3.4.)

quantum transport in 1-d complicated structures, we assume neither artificial hierarchical (e.g., Fibonacci-type) nor random potentials. Let us write the wavefunction

as a sum of the incident and reflected waves for xFc) Tn shows a sequence of steadilyelongated plateaus that are connected by bursts. (See Fig. 3.3(a).) Both

Chapter 3

36

f

Fig. 3.2. Transport properties of -function model for F =0.001: (a) Tn, (b) S 11(n), (c) power spectrum P(f) of (a) in logarithmic scales, including a reference line corresponding to P f2.

the direction (upwards or downwards) and the magnitude of each burst are unpredictable, which is in marked contrast with the feature of the intermittent chaos, where only the location of bursts is erratic.

Pseudo- Chaos

37

Fig. 3.3. Same as in Fig. 3.2, but for F =10.

The component S 11 (n) in Fig. 3.3(b) wanders from one stagnant region to another in an unpredictable way, whose overall feature looks like a living animal. As in the case (I), however, the positive Lyapunov exponent is vanishing. The power spectrum of Fig. 3.3(a) again shows the 1/f 2 law (Fig. 3.3(c)). Locally, a picture of the tilted band structure

38

Chapter 3

is meaningful. Laminar oscillations in the plateaus (see the inset of Fig. 3.3(a)) are caused by the increase of kn values within each band, whereas bursts are due to the Zener tunneling to adjacent bands at the zone boundaries. In fact, we recognize that the burst occurs regularly whenever k n=k BZ be satisfied, which is also distinct from the feature of ordinary intermittent chaos. We can further proceed to examine the transport properties in generic systems with barriers of finite width for the cases of d/ =0.1 and 0.5. In this instance the problem is actually the over-barrier Despite this fact, a gross feature of Tn for F =10 has transmission. proved identical to Fig. 3.3(a) and the corresponding power spectrum obeys the 1/f 2 law. Thus the integer exponent in the 1/f 2 law holds universally for periodic superlattices in the electric field, irrespective of the field strength and the width of barriers. To characterize the nonstationary behavior of Tn, we shall calculate the Allan variance (Allan, 1966; Mandelbrot, 1968; Aizawa et al., 1989) (3.3) In the Markovian process, the variance in (3.3) tends to zero as N is y increased, satisfying a scaling law A2 = N with y V0 denote the height of barriers of b1,b2 and B1, B2 , respectively. The semi-infinite barriers serve to confine the electrons. The term n(x;A) is a characteristic function defined by (3.5) In a mean-field approximation, one may resort to a Hartree-like equation describing the motion of the electron cloud. In terms of a (x,t), the dimensionless charge inside the single-body wavefunction well is given by (3.6) We then equation

obtain

the

time-dependent

(single-body) Schrödinger

(3.7)

Pseudo- Chaos

41

which is obviously nonlinear with respect to The coupling constant measures the strength of the mean field acting on each electron. As an initial state, let us choose the wave packet located inside the well w1 and moving towards the well w 2 with mean momentum : .

(3.8)

If /w1N with N correspond to propagating and evanescent waves, respectively. In the cavity region, on the other hand, the Green function is given by (Feynman and Hibbs, 1965; Ueta, 1992)

(4.9) where w is a scaled energy and W is the Whittaker function of To determine the values for S matrix, we exploit Green's theorem which yields the equation including integrations along the closed boundary C of the cavity region:

(4.10) where r and r' lie on C and signifies the outward-normal derivative on C; and (r) denote Cauchy's principal value and the interior angle at r, respectively. In (4.10), no contribution from the bulk integration arises, since r also locates along C. We shall here apply the boundary element method: The boundary of the cavity region is approximated by a sequence of small line segments. Then functions in the integrand along each integration segment in (4.10) are approximated by their linear interpolations: For a segment connecting ri and ri+1, for instance,

(4.1la)

Chaos and Quantum Transport

57

where (4.1 1b) 1. Substituting into (4.10) G(r, r'; ) from (4.9), its normal with -1 derivative, and expressions from (4.11), we find (4.12) where J, P and Q are numerical coefficients and the boundary condition rL+1 =r1 is imposed. Among unknown variables and =0 is satisfied at the wall and and at the holes j=1 and 2 are rewritten in terms of S matrix as:

(4.13b) Using these notions in (4.12), we eventually obtain a set of linear equations for unknown variables S(1) , S(2), and wall , whose solutions lead to the flux-normalized transmission coefficient tmn = S(2) mn and the magneto-conductance g(B)=(2e2 /h) Similarly, wavefunctions are available by substituting the solution into (4.10) with r taken inside the cavity region for S(j) and wall together with a new choice (r) =2 Semiclassical Treatment Before proceeding to the numerical analysis, we shall mention a way of deriving the semiclassical formula for S matrix. In open billiards, a point particle escapes to exit wires after being temporarily confined within the cavity, which is distinct from situations in closed billiards. Owing to this fact, one will be free from the problem of the divergence (i.e., exponential proliferation coming from summation over infinitelylong periodic orbits) proper to Gutzwiller's semiclassical trace formula (1990). Since our major concern in this chapter lies in the quantummechanical explanation of the experimental results, the semiclassical

58

Chapter 4

approach will be briefly presented below. Suppose an electron enter in mode n from the wire 1 and escape in mode m to the wire 2. The quantum-mechanical expression for a transmission element of the S matrix is then given by (Baranger et al., 1991; 1993a,b)

(4.14) where y' and y" are coordinates at the entrance and exit, respectively, on the boundary C, with corresponding x coordinates fixed as x'= and x''= denote transverse The symbols { (y)} and G(y ‘’, y') components of the wavefunctions at junction points (see (4.8)) and the Green function through the cavity, respectively. Finally, vn(v’) and vm(v’’) are parallel (to wires) components of velocities for modes n and m, respectively. The semiclassical 0) approximation to (4.14) will be made as follows: We first replace G(y’’, y') by the semiclassical Green function:

(4.15) where the summation is taken over all the classical orbits starting at y' and ending at y''. In (4.15), S j J

p• dq) ,

(= det

|v’v’’| ), and 2 µ j / are reduced action, inverse of the Jacobian, and Morse-Maslov index, respectively. Using (4.15) in (4.14) and evaluating the double integrals in (4.14) by a saddle-point method, we reach the semiclassical result

j

(4.16) -½

where D j = and with = ± m; and are length of orbit j and Fermi wavelength, respectively. The summation in (4.16) is taken over all the isolated open orbits satisfying the . saddle-point condition (py For the actual computation of (4.16), one should vary the value y' and have an assembly of orbits that satisfy py' = h and py'' = at the entrance and exit, respectively. The Poincare' map in (4.1) will be a convenient tool for obtaining these orbits. After finding orbits, we


59

should resort to the linearized Poincare' map connecting between variations at the initial and final points (see (4.3) and (4.4)), obtaining the derivatives as . For instance, noting =0, at the entrance and at the exit, one finds Dj= results will be used as input in (4.16). A further details will be omitted here.

All these explanation of

4.5. Comparison in Stadium Billiard between Theory and Experiment We shall now embark upon describing the numerical issue derived from quantum theory and interpreting Marcus et al. 's experimental results. Assume an injection of a single-mode (N=1) by choosing =1.2. Taking B0=(hc/e)/A as a unit of the magnetic field, the scaled magnetic field B /B0 will be varied between 1.72 and 27.62. In terms of Larmor radius R(= this regime corresponds to 0.397B0, the ergodic part begins to be replaced successively by the KAM tori and thereby the phase space shows an extreme sensitivity to the variation of B, which explains a rapid variation of g(B). To characterize the fluctuation of g(B), the autocorrelation functions have been computed for both the low and high reference fields (see Fig. 4.7); means the average over referenced B fields. Both Cl and Sd billiards are commonly accompanied by the cusp-like central peaks proper to


61

Fig. 4.7. Autocorrelation functions Reference ranges are 00. Confining ourselves to the A1 representation throughout the present chapter, the resonance of the S matrix are determined by (5.10) in the complex plane of the wave number We shall examine the ranges with and for the ratios and (A1) amounts to a dimension respectively. The size of the matrix M of 50x50, for us to obtain well-converged values of its (scaled) determinant for any fixed value. The computed results are given in Figs. 5.3 and 5.4 for the case of and 12, respectively. These figures will be used for comparison with the semiclassical results in the next section.

5.3. Semiclassical Theory Let us define a relative density of states ρ (E) as the difference between the state densities of free and scattering systems, both confined in the

86

Chapter 5

Fig. 5.3. Exact quantum ( ) and semiclassical( x ) resonances in case of Semiclassical results using: (a) 6 orbits with periods 1 an 2, (b) 14 orbits with periods 1,2, and 3.

Fig. 5.4. Same as in Fig. 5.3 but orbits with periods 1, 2, and 3.

=12. The semiclassical result consists of

Chaotic Scattering on Hyperbolic Billiards very large circle of the radius of 1974)

87

. The function ρ (E) is asymptotically independent and is related to the S matrix as (Balian and Bloch,

(5.11) Therefore, the density ρ (E) and the S-matrix share the same poles. The -1 Im g(E) , with g(E) being function p (E) itself is written as ρ (E) = the trace of the difference between the Green functions with and without the four disks. In the semiclassical limit, g(E) is expressed as a sum over periodic orbits, i.e., by Gutzwiller's trace formula:

(5.12) where g0 is E-independent and given by the difference between Thomas-Fermi state densities with and without four disks:

Here p and r denote any unstable and isolated prime periodic orbit and the number of its repetition, respectively. With signifying the = and Tp(E) = length of the periodic orbit, Sp(E) = dSp(E)/dE = are the reduced action and the period, respectively. (For convenience, we rescale the velocity to a unit value v=1.) Lp is the number of collisions of the orbit p with disks. The stability via exponent µ p is derived from the stability eigenvalue is the larger (The Lyapunov exponent λp, is given by , eigenvalue of the monodromy matrix Mp constructed from the linearized Poincare' map for coordinates perpendicular to the trajectory. The advantage of the present system lies in the E independency of due to the constancy of the particle velocity (v=l), which greatly simplifies the summation in (5.12). Equations (5.11) and (5.12) indicate that the scattering resonances can be obtained from poles of g(E). With a help of symbolic codings, we can get all the periodic orbits

88

Chapter 5

Table 1. Periodic orbits up to period 3. The 3rd column represents a sequence of disk numbers in the order of bouncings.

Chaotic Scattering on Hyperbolic Billiards

89

without missing any one of them, pictures of 14 periodic orbits up to the period 3 in the number of bounces being listed in Table 1. As pointed out in Chap. 2, the trace formula in (5.12) is not convergent as it stands, i.e., conditionally convergent. Using the expansion

one can write the trace function in terms of the Ruelle zeta function as (5.13a) with

(5.13b)

where =exp(µ p). The form of product sum in (5.13) guarantees the better convergence. Finally, the complex poles of the Ruelle zeta function yield resonances of the S matrix. Noting the C4v symmetry of the system, the function can be factorized as , with each factor corresponding to the irreducible representations A 1, A 2, B1, B 2 and E. This symmetry-induced factorization further simplifies the procedure of taking the product over periodic orbits in (5.13b). We have, for instance (Cvitanovic and Eckhardt, 1989, 1993),

(5.14a) where tp is given by (5.14b) so long as we restrict ourselves to the resonances with the longest lifetimes, which are given by the first Ruelle zeta function with j=0. How can we compute and necessary to evaluate tp in (5.14) ?

90

Chapter 5

The answer is as follows: Let be coordinates on the Poincare' surface of section for the periodic orbit p and then derive the Poincare' map for the flow in the phase space. By linearizing this map, we obtain the map for variations of the orbit p:

The eigenvalue of the monodromy matrix M is nothing but the stability eigenvalue which are (5.15) [Note that up below (5.12) is

and the Lyapunov exponent is

The explicit form of M is available from a simple geometrical inspection as

(5.16) where and are the length of the flight between disks and the reflection angle, respectively, and are found from the profile of periodic orbits in Table 1. The quantity (=1 for all n) is the curvature of the disks where a particle bounces. For illustration, let us choose the n=l orbit "0." Then, =1 and leading to and with the degree of opening . As a little more complicated example, we consider the n=3 orbit "022" (see Fig. 5.5). By application of Pythagoras' theorem and the sine formula to right-angled triangles, OPP' and OPP", respectively, one gets


91

Fig. 5.5. Flight distances and incident angles of period-3 orbit "022" in Table 1.

By solving these equations simultaneously, values of y and x=cos are determined, and other variables are given by cos( )=1, and I31 =y. Substituting cos( )=cos( )=x, l12 =l31 =y- I23 =R- x these results into (5.15) and (5.16), we obtain both and For other periodic orbits in Table 1, similar calculations as above are possible. Finally, with use of the values and , tp in (5.14) is determined for any periodic orbit p. It should be noted that the Bunimovich-Sinai curvature formula (1980) can also be used to obtain the Lyapunov exponents. We are concerned with the A1 representation which proves to include the scattering resonance with the longest lifetime bordering a “gap” void of resonances formed just below the real axis. The values satisfying resonances for the A1 sector are given by complex (5.17)

92

Chapter 5

In a quantum-mechanical treatment, the computation of scattering resonances demands an excessively long cpu time and a large region size for storage of memories. By contrast, the computation of (5.14) and (5.17) is not laborious, and one may promptly get a large number of resonances by which one is able to analyze their statistical properties. For the present system, therefore, the semiclassical theory is superior to the exact quantum theory.

5.4. Distribution of Complex Resonances The location of poles, i.e., of complex resonances are given in Figs. 5.3 and 5.4 in the cases =6 and 12, respectively. While the resonances { } lie on the real axis in closed systems, they have nonvanishing imaginary components in open scattering systems. The semiclassical resonances based on 6 orbits with periods 1 and 2 and those based on 14 orbits with periods 1, 2, and 3 are displayed in Figs. 5.3(a) and 5.3(b), respectively. From Fig. 5.3(b), we learn the following: (i) In a wide window of 0>1 where diffraction effects can be neglected. In the case of crossroad, we see that the shortest reaction times are obtained for the smallest values of the corner radius a, assuming a fixed value for the width (=R-2 of the lead wires. For GaAs/AlGaAs heterojunctions, the effective mass of the electrons is m=0.067m e , and an electron density of ns=3 x 1011cm-2 can be obtained so that the Fermi velocity would then take the value m=2.4 x x105m/s. For a nanometric circuit of size R=100nm, the time unit is therefore of R/uF=0.4x 10-12s so that the lifetimes of the resonances are in the subpicosecond domain. We suggest that the dynamical behavior of such devices could be probed by femto-second laser experiments. To conclude, the semiclassical theory on chaotic scattering on convex hard disk systems has the following advantages: (1) The systematic enumeration of all the periodic orbits is possible owing to the symbolic dynamics. (2) So long as one is concerned with the case of the large degree of opening (i.e., of filamentary disks), only short periodic orbits contribute substantially to the trace formula, eliminating the problem of its divergence due to exponential proliferations. (3) Since poles of g(E) and S(E) are identical, their imaginary components are significant, while they are not demanded in bounded systems. As a consequence, the semiclassical theory of chaotic scattering has not only a conceptual significance to uncover the quantum-classical correspondence, but also an advantage to be more practical than the exact quantum theory which, in applications, meets computational limitations in both cpu time and memory area.

5.5. Experiment on Antidot Arrays in Magnetic Field While effects of magnetic field have not been taken into consideration in the theoretical treatment above, recent experiments have accumulated around the quantum transport in regular arrays of hard disk billiards in the presence of magnetic field. Weiss et al. (1991, 1993), using high-mobility GaAs/AlGaAs heterojunctions having carrier density

98

Chapter 5

n8~2 x x1011 a" and mobility µ -1.0 x 106 cm2 /Vs, fabricated a periodic square antidot array (: periodically-drilled holes through a twodimensional electron gas) with the period R=200~300 nm (which of each should not be confused with Rc below) and with radius antidot (electron depletion region) less than R/2. Applying a magnetic field B perpendicularly, they explored transport anomalies in a temperature region where both the electron mean free path le and the phase coherence length are of the order of 10 µ m. Note the classical cyclotron radius at the Fermi energy. The measurement of resistivity pxx for a sample with R~0.5 at T~0.4K as a function of B shows distinct quantum oscillations in the low B regime (2R c>Rwhich differ from the Shubnikov-de Haas oscillations periodic in 1/B in the high B regime (see Fig. 5.8). The latter is caused by the Landau level structures. Since le, Iφ >>R and the Fermi wavelength ~ 50nm, the quantization of chaotic motion of electrons plays a role. The quantity ρxx in Fig. 5.8 displays quantum oscillations superimposed upon the low-B resistance anomalies. These oscillations are periodic in B with period B~ 0.105 T [Tesla]~h/eR2 . The crossover from Bperiodic to 1/B-periodic oscillations, together with the fact that the antidot array is much larger than the phase-breaking length in a 2DEG, suggest that the B-periodic oscillations are not due to the Aharonov-Bohm interference of phase-shifted wave functions. Assuming ρxx~g2(E, B), Weiss et al. tried to explain the experimental issue by resorting to Gutzwiller's trace formula. This system, though simulating the so-called Sinai billiard, is a generic and mixed system since electrons are subject to the magnetic field. Furthermore, not a hard but rather soft wall model is employed in their study. As a consequence both stable and unstable periodic orbits contribute to g(E, B). The relevant periodic orbits are displayed in Fig. 5.8(b). Due to finite temperatures and impurity broadening effects, only the shortest periodic orbits are taken into consideration. The periodic orbits cause a modulation of g(E, B ) with maxima given by the quantization rule (5.21) where N=(n+y/2+ /4) in terms of the quantum number n, the winding number , and the Maslov index α. B B) is the flux penetrating the area B) enclosed by a periodic orbit. For comparison with experiment,


99

Fig. 5.8. (a) ρxx measured in the patterned (top traces) and unpatterned (bottom trace) segments of the same sample for T=0.4K (solid lines) and 4.7K (dashed line). Inset: ρxx up to 10 T with filling factor v =2 marked by arrow. (b) Triangles mark 1/B positions of pxx minima. Solid, dashed, and dotted lines are calculated reduced actions (1 /B) of orbits (a), (b), and (c), respectively. These orbits are shown for 1 /B=0.6T-1 (top) and 1 /B= 2.7 T-1 (bottom inset). (Courtesy of D. Weiss et al. )

the reduced action

is more advantageous: S(B)=2n with n=1,2,• • • labels minima of

100

Chapter 5

g(E,B). Three periodic orbits in Fig. 5.8(b) are sufficient to explain the minima of pxx . These are: (a) an orbit between four antidots, (b) another one around a single antidot, (c) the last one emerging from a bifurcation of orbit (b). All other periodic orbits are asserted to play only minor roles. The calculated (B) traces in Fig. 5.86) highlight the dominant role of the orbits (a)-(c). Despite an elegant interpretation of some of the fluctuations in quantum transport, there remains an open question concerning the underlying low-B field anomaly, which should be interpreted on the basis of isolated unstable periodic orbits. Specifically, it is possible to have to give alternative insight into the fluctuation in the vicinity of zero field when the system is fully chaotic and no stable periodic orbit survives. It is therefore highly desirable to extend the theory given in the present chapter to the system with more than four disks and also in the presence of magnetic field.

References Alonso, D., and Gaspard, P. (1994). J. Phys. A27, 1599. Balian, R., and Bloch, C. (1974). Ann. Phys. 85, 514. Beenakker, C. W., and van Houten, H. (1991). In Solid State Physics: Advances in Research and Applications, H. Ehrenreich and D. Turnbull, eds. New York: Academic. Bunimovich, L. A., and Sinai, Ya. G. (1980). Commun. Math. Phys. 78, 247,479. Cvitanovic, P., and Eckhardt, B. (1989). Phys. Rev. Lett. 63, 829. Cvitanovic, P., and Eckhardt, B. (1993). Nonlinearity 6, 277. Gaspard, P., and Alonso, D., Okuda, T., and Nakamura, K. (1994). Phys. Rev. E50, 2591. Gaspard, P., and Nicolis, G. (1990). Phys. Rev. Lett. 65, 1693. Gaspard, P., and Rice, S. A. (1989). J. Chem. Phys. 90, 2225, 2242, 2255 (1989). Gaspard, P., Nakamura, K., and Rice, S. A. (1991). Comments Atom. Mol. Phys. 25, 321. Doron, E., Smilansky, U., and Frenkel, A. (1991). Physica D50, 367. Eckmann, J. P., and Ruelle, D. (1985). Rev. Mod. Phys. 57, 617. Giannoni, M. J., Voros, A., and Zinn-Justin, J., eds. (1991). Chaos and Quantum Physics, Proceedings of the NATO ASI Les Houches Summer School. Amsterdam: North-Holland.


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Gutzwiller, M. C. (1990). Chaos in Classical and Quantum Mechanics. Berlin: Springer. Jessen, B., and Tornehave, H. (1945). Acta Math. 77, 137. Krylov, N. S. (1979). Works on the Foundations of Statistical Physics. Princeton: Princeton University Press. Lax, P. D., and Phillips, R. S. (1967). Scattering Theory. New York: Academic. Miller, W. H. (1975). J. Chem. Phys. 63, 996. Nakamura, K. (1993). Quantum Chaos: A New Paradigm of Nonlinear Dynamics. Cambridge: Cambridge University Press. Porter, C.E. (1965). Statistical Theories of Spectra: Fluctuations. New York: Academic. Porter, C. E., and Thomas, R. G. (1956). Phys. Rev. 104,483. Rice, S. A., Gaspard, P., and Nakamura, K. (1992). In Advances in Classical Trajectory Methods 1, W. L. Hase, eds. Connecticut: JAI Press. Ruelle, D. (1978). Thermodynamic Formalism. Reading, Massachusetts: Addison-Wesle y. Ruelle, D. (1986). Phys. Rev. Lett. 56, 405. Walters, P. (1981). An Introduction to Ergodic Theory. Berlin: Springer. Weiss, D. et al. (1991). Phys. Rev. Lett. 66, 2790. Weiss, D. et al. (1993). Phys. Rev. Lett. 70, 4118.

Chapter 6 Nonadiabaticity-Induced Quantum Chaos To accommodate temporal chaos in quantum dynamics, we consider the dynamics beyond the Born-Oppenheimer approximation. After reviewing the fictitious gauge strucure in the adiabatic limit, we examine the role of gauge structure in nonadiabatic transitions for transport in open paths. Local features of the gauge potential modify the nature of the intersection of the adiabatic energy surfaces and thereby affect crucially the Landau-Zener formula for a single-passage transition rate. We then investigate the full dynamics of systems involving both slow (nuclear) and first (electronic) dynamical variables and demonstrate the chaotic behaviors in quantum dynamics.

6.1. Avoided Level Crossings and Gauge Structure As we have so far insisted, the collapse of tori and onset of chaos are disturbing the foundation of the formalism of quantum mechanics in the adiabatic limit, i.e., when no quantum transition occurs. In this limit the present formalism of quantum mechanics demonstrates many puzzling features: avoided level crossing and gauge structure, commonly appearing at the level degeneracy, are the most interesting events, which we shall explain below. The adiabatic limit correponds to stationary states, in which the major problem of quantum mechanics is to solve the eigenvalue equations 102

Nonadiabaticity-Induced Quantum Chaos

103 (6.1)

where in general the Hamiltonian H(q) depends on parameters R=(R 1 , R 2, • • •, R d) such as external electric and magnetic fields. Within a desymmetrized manifold, no degeneracy can be seen except for the accidental ones (von Neumann and Wigner, 1929): If a single parameter were varied, we would see only level repulsions, e.g., avoided level crossings. We shall illustrate this point by using a two-states model. and are degenerate Suppose that at some point R0 two states with common energy E0. The eigenvalue problem in the vicinity of R0 can be solved by degenerate perturbation theory. Using as diabatic (R-independent) bases and I the matrix elements of H(R) are given as (6.2) with H'ij(R) =

The energy splitting at R is (6.3)

To meet the degeneracy (or to make E =0), both equalities (6.4a) and (6.4b) should be satisfied. The number s of independent equations in (6.4a) and (6.4b) is 2, 3, and 5, respectively, for real symmetric, complex Hermitian and quaternionic Hamiltonians. Hence, if d<s, no solution for R exists satisfying (6.4), and one cannot see any degeneracy except for accidental ones. For d=s we can have a set of isolated degenerate points. Finally, for d>s the degeneracy is not isolated, lying on a g(=d-s) -dimensional manifold, e.g., lines for g =1, surfaces for g =2, etc. This situation has been extensively studied by Arnold (1978) in the context of the theory on a surface of the second order (quadric): The condition for degeneracy to occur is identical to that for a given ellipsoid to become the ellipsoid of revolution. In fully chaotic systems, we have no integral of motion other than the total energy. The corresponding quantum systems are therefore

104

Chapter 6

desymmetrized at the outset. When a single quantity is vaned among d( 2) nonintegrability parameters, we shall see only avoided level crossings. In the semiclassical limit the level density is very high, and these avoided crossings constitute the backbone of complicated energy spectra. Below we shall examine in detail a structure of the vicinity of the avoided crossing in case of d 2. The eigenvalue problem (6.1) is closely related to quantum dynamics in the adiabatic limit that R changes very slowly as a function of time t. Let us find the corresponding adiabatic change of the electronic wavefunction I lying initially at by investigating the time-dependent Schrödinger equation (6.5) In the adiabatic limit, one may regard R as frozen at each moment in time. By exploiting (6.1), the solution of (6.5) is given by (6.6) where the phase that the wavefunction acquires in its adiabatic change, is defined by

(6.7) In the case d 2, is a path-dependent geometric phase that cannot be vanishing in general (Berry, 1984), while in the case d=1 it can always be smeared out by suitably after choosing the phase of This is a manifestation of the fact that eigenstates are multivalued in the parameter space R in d( 2) dimensions, though they are single-valued in configuration space. Let us consider a closed circuit C in R space (see Fig. 6.1) and introduce the fictitious gauge potential (6.8)


105

For an adiabatic transport of the wavefunction along C, the geometric phase is now expressed as

.9) where dS denotes an area element of the surface enclosed by C, and Vn (R) is a fictitious magnetic field defined by

0) [The vector products in (6.9) and (6.10) follow Berry's paper (1984). They should be rewritten in general in terms of exterior differentials.] The form (6.10) indicates that Vn(R) is singular at degenerate points. For the case d( 2), the parametric change of an energy eigenvalue

Fig. 6.1. Conical intersection of a pair of energy surfaces and closed path C.

106

Chapter 6

constitutes not a line but a surface. The adjacent energy surfaces is topologically fictitious gauge potential. As a concrete example, let us take magnetic field (R=(X, Y, Z )) described by Hamiltonian

crossing point of a pair of singular and generating the a model of a spin in the an SU(2) complex Hermitian

(6.11) where I and denote the unit matrix and Pauli matrices, respectively. In this example, d=s=3. A pair of energy surfaces (eigenvalues) are represented by (6.12a) and they are conically intersected at the diabolical point R=0 (see Fig. 6.1). In terms of polar coordinates defined by R=(sin cos sin sin cos ), the corresponding eigenstates are given by (6.12b) where have

Substituting (6.12) into (6.10), we

(6.13) This is equivalent to a magnetic field induced by a monopole at R=0 with charge ± 1/2. Using (6.13) in (6.9), the geometric phase is given by (C) = where is the solid angle for the view of circuit C as seen from the degeneracy. When C encloses (with any plane circle including R=0) the degeneracy, = ± 2 otherwise, =0. In the former case, exp (C)} =-1, which changes the sign of the eigenstates. The phase (6.9) is a manifestation of a holonomy (i.e., failure of some variables to return to their original values after a cyclic change of other variables) for the parallel transport of quantum eigenstates (unit vectors) in parameter space (see Fig. 6.2). In the field of quantum chemistry, vibration-rotation energies of molecules are analyzed in the Born-Oppenheimer approximation, wherein electronic quantum states are assumed to follow the change of nuclear coordinates

Nonadiabaticity- Induced Quantum Chaos

Fig. 6.2. Parallel transport of state vector along spherical surface. Path: 1 3 4.

107

2

adiabatically. Longuet-Higgins et al. (1958) noticed the geometricphase-induced sign change in the electronic wavefunctions when nuclear coordinates are cycled. In the field of differential geometry, Darboux (1896) had found a similar sign change in his analysis of "umbilic points" of curved surfaces. Nevertheless, the universal gauge structure around level degeneracies was not revealed until Berry's work.

6.2. Nonadiabatic Transitions and Gauge Structure The discovery of quantum adiabatic phase accompanying transport along closed paths has indeed had a great impact on various fields of physics and chemistry (Shapere and Wilczek, 1989). This global phase, originally discussed in connection with the intersection of molecular energy surfaces, is attributed to connections in a Hilbert bundle, i.e., to a fictitious gauge potential. This gauge structure is a consequence of the novel complex nature of the wavefunction in the time-dependent Schrödinger equation. A generalization of the influence of the quantum adiabatic phase to

108

Chapter 6

the nonadiabatic transitions is nontrivial, because we capture the local (rather than global) structure of the novel gauge potential via the transition rate. While several studies are concerned with this theme, most of the efforts have been concerned with corrections to the global geometric phase factor for closed or near-closed paths (Aharonov and Anandan, 1987; Moor, 1991; Wu and Kuppermann, 1993). Nonadiabatic transitions, however, occur widely for both open and closed paths and they provide a key to understanding a variety of state-changing phenomena, e.g., two-level dynamics in the presence of magnetic and/or electric fields, atomic and molecular collisions, Zener tunneling, etc. (Dykhne, 1962; Mott and Massey, 1965; Pechukas, 1969; Miller and George, 1972; Child, 1974; Hwang and Pechukas, 1977; Nikitin and Umanskii, 1984). This kind of transition is induced at the avoided crossings of the potential surfaces. In this section, we consider nonadiabatic transitions between a pair of states as a mechanism for nonadiabatic transport along open paths. In particular, following Nakamura and Rice (1994), we show the influence of the fictitious gauge potential on the transition rate. Although we use, as a prototype, the dynamics of a single spin in the presence of a time-dependent magnetic field, the present analysis is applicable to other systems, e.g., two-level systems subjected to timedependent laser fields. In the magnetic field B(t) = (Bx(t), By(t), Bll(t)), the spin dynamics is described by the Schrödinger equation, with (6.14) where the negative of the Bohr magneton µ B has been suppressed for and By=0, we obtain the simplicity. With the choice Bll =vt, Bx= Landau-Zener (or curve-crossing) model. Here both Bx and By are assumed nonvanishing and time-dependent: B(t) executes a winding (besides a translational) motion. We shall evaluate the transition amplitude from one adiabatic state at t =to another at t =+ . First we briefly summarize the existing formula (Berry, 1990) for the transition rate in the presence of winding motion. With the use of adiabatic bases (t) and (t), (t) is written as (t) = C1(t) (t) + C2(t) (t) and the Schrödinger equation is reduced to


109

(6.15) with (6.16) where overdots denote time derivative, the {Ei } are the eigenvalues of H in (6.14) at particular times (adiabatic energies), and the {Ai} are (mutually different) fictitious gauge potentials. In our notation, i =1 and 2 are associated with the lower and higher energy states, respectively. This gauge structure leads to a novel geometric phase for adiabatic transport along closed paths. As for nonadiabatic transport, use of the path-integral method and the stationary phase approximation leads to the following transition rate for a single passage of the avoided crossing: p=exp(-2 with (6.17) where E = E2-E1, A = A2-A1, and tc is the complex crossing point at which E vanishes (Berry, 1990). According to the rigorous treatment given below, however, the gauge potentials enter into the phase integral in a much more intricate way, and (6.17) is valid only in a very special limit. We now examine the quantum dynamics for the Hamiltonian (6.14) using a diabatic (time-independent) basis. Transforming to polar coordinates defined by and introducing the unitary transformation

we introduce a new wavefunction

defined by

Then

satisfies (6.18)

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The transformed Hamiltonian given by (6.18) is now analytic throughout some strip of the complex-time plane centered on the real axis, and it satisfies I for (i.e., absence of mixing between diabatic states). In this situation we can apply the phase-integral method (Stückelberg, 1932; Crothers, 1971), obtaining for the full transition rate

(We are not interested in with the single-passage rate p = exp(-2 the phase-interference factor, sin2(• • •), proper to the passage of successive avoided crossings.) is now given by (6.19) where (6.20) i.e., the difference between adiabatic energies of in (6.18). Note that is the complex crossing point nearest to real axis. Equations (6.19) and (6.20) are exact. In the near-adiabatic region is constant and of the winding Landau-Zener model, predominant, whereas and Bll are slow and small variables in the transition region near the avoided crossing. In this extreme case, (6.20) is expanded in one of the slowest variables as

1a) and (6.21b) The corresponding expansion of will follow in the integration (6.19). While the results in (6.19) - (6.21) are derived using the diabatic basis, we now rewrite them in terms of the adiabatic basis: Definition and with E (6.16) yields A =A2 - A1 = = Using these expressions in (6.20), we have (6.22)


111

With the same replacement in (6.21), the asymptotic behavior is given by (6.23a) (6.238) with approximated by the values at which E and vanish in (6.23a) and (6.23b), respectively. Therefore (6.17) is justified in the special limit when the winding motion is much slower than the translational motion. In the opposite limit, the sum of gauge potentials is needed and, in the general case, we should use (6.22). We shall now apply the results in (6.19), (6.22), and (6.23) to a winding Landau-Zener (curve crossing) model where Bll = vt, Bx = and By = with = wtn/n (n=1,2,• • • ) . For simplicity, v, and w are assumed positive. Below we shall use E = + (vt)2]½ , irrespective of the value of n. Case of n=1. In this case, the Galilean transformation (t t'=tw/v) smears out the phase factor, so that there is no gauge structure is and the ordinary Landau-Zener transition rate pLZ = applicable Case of n=2. Using in (6.23) A = and A = wt, we find, from (6.19), (6.24a) (6.24b) With the same expression for gauge potentials in (6.22), (6.19) with yields = which recovers the two limits in (6.24). While Berry carefully analyzed the n=2 case, he was mainly concerned with (6.23a) and (6.24a), showing neither (6.22) nor (6.23b). Case of n=3. In this case, A = wvt3 / E and A = wt2 . The formula in (6.17) or (6.23a) here yields no correction to the LandauZener's result, since the integration of A does not have any imaginary component. On the other hand, from (6.23b) we have =0.61 x In terms of a scaled time the most general result available from (6.22) is

112

Chapter 6

with ε ~ ~ 0.61-0.42

and = + for ε 1, and

(6.25a) and (6.25b) for 4 and respectively. The asymptotic behaviors in (6.25) are in excellent agreement with an envelope of the full transition rate in Fig. 6.3 obtained by our numerical iteration of the time-dependent Schrödinger equation (6.14) for n=3. (Spiking oscillations in P are attributed to the multiplicative phase-interference factor in the equation above (6.19).)

Fig. 6.3. and =1. (a) represent arrow in

Full transition rate P for the winding Landau-Zener model with =1 Filled circles denote values obtained by numerical iteration of (6.14): with v=0.4, (b) with v=0. Envelope lines in (a) and (b) with p in (6.25a) and (6.25b), respectively. The Fig. 6.3(a) corresponds to the limit p=pLZ.


113

We here assert: (i) Even when the winding motion is slower than the translational motion, the existing formula (6.17) fails in providing a leading-order correction to pLZ. (ii) In the opposite case, the formula in (6.23b), consisting only of a sum of gauge potentials, can give accurate leading-order values for the transition rate. (iii) In general, (6.22) indicates the transition rate depends on the difference and sum of a pair of gauge potentials in intricate ways, according to the adiabaticity ratio of the winding and translational motions. The present formula in (6.22) and (6.23), with a slight modification, can also be used in other winding models in the near-adiabatic region. Consider, for instance, a winding Demkov (exponential) model = and = (Demkov, 1964) with Bll = sin exp(-vt) in the case Interchanging Bll and B the formulas from (6.21) through (6.23) still hold; this interchange comes from the predominance of Bll in this model. Noting the gauge potentials

and with

we have, from (6.23) and (6.22), (6.26a) (6.26b) and (6.27) [Due to a finite mixing of respectively, where pDemk = diabatic states in the limit the exact expression for a single passage transition rate in the winding Demkov model is p=(1+ )-1, which, however, reduces to a standard form p=e in a near-adiabatic regime.] The winding Demkov model, after a unitary transformation (6.18), turns out the exactly solvable Nikitin's model (1970), whose known solution justifies (6.26) and (6.27) in the near-adiabatic region. In conclusion, when the applied magnetic and/or electric fields show

114

Chapter 6

time dependence in both amplitude and polarization vector, the local aspect of the gauge potentials greatly affects the intersection of adiabatic energy surfaces. The formula for nonadiabatic transition rate includes a pair of gauge potentials in intricate ways, depending on the ratio of the adiabaticities of the winding and translational motions of applied fields. Experimental Evidence The interference in transitions due to local geometric phases has been observed by Bouwmeester et al. (1996). They chose an optical system in which the polarization dynamics of the light can be described by a Schrödinger-like equation. Two optical levels are formed by two orthogonal polarization eigenstates of a single longitudinal mode of a ring cavity. The two adiabatic eigenstates are determined by three birefringent elements in the form of electro-optic modulators (EOMs) placed inside the optical cavity. The birefringences form X, Y, and Z coordinates of the parameter space and are controlled by time-dependent electric voltages applied to EOMs. The cavity decay time (~µ s) is long enough to complete the whole process of measurement. The dynamics

Fig. 6.4. Three pairs of branching points that dominate the transition phenomena for Gaussian twisted Landau-Zener model. =0.44MHz/µ s, =0.33MHz, =1 µ s and µ =5.4 rad. These parameter values are also used in Fig. 6.5. (Courtesy of D. Bouwmeester et al. )


115

of the intracavity field is measured by analyzing the polarization of light that leaks out through the cavity mirrors. Bouwmeester et al. have examined the case of Gaussian twisted Landau-Zener (LZ) model by taking in (6.18) and the phase function The transition probability, being dominated by three pairs of branch points (see Fig. 6.4), is expected to show interference phenomena between successive avoided crossings. The approximately analytical expression for the transition probability P as a function of the adiabatic parameter can be obtained by using the methodology described so far. The most essential issue is the presence of a local minimum in P for the twisted LZ model, which is a clear indication of interference between the branch points. Figure 6.5 shows the time trace of the normalized intensity of +

Time (µs) Fig. 6.5. Experimental results: curve 1 shows the time trace of for the conventional Landau-Zener model with =0.44MHz/µ s, =0. 33MHz; curve 2 shows the time trace for Gaussian twisted Landau-Zener model. Dotted curves are the corresponding traces obtained by numerical simulations. Vertical dotted lines indicate the time positions corresponding to real parts of branching points in Fig. 6.4. (Courtesy of D. Bouwmeester et al.)

116

Chapter 6

polarization Curve 1 is the experimental result of the ordinary LZ model in the near-adiabatic region. The oscillation after the avoided crossing at t=0 is caused by the strong mixing between + and - polarizations (i.e., a pair of adiabatic eigenstates) in the crossing region. Curve 2 is for the Gaussian twisted LZ model, indicating the influence of the three pairs of branch points in Fig. 6.4. In fact the intensity shows radical changes at t=0 and t=± 0.93 µ s which are just the real parts of the branch points in Fig. 6.4. The dotted curve is a numerically computed one which is in good agreement with the experimental issue. Figure 6.6 shows the final transition probability P between the adiabatic energy levels as a function of the adiabatic parameter Λ. A good agreement between the experimental data and numerical and approximate-analytical results. The key point is the presence of the

Fig. 6.6. Square points are experimental values for final transition amplitude P as a function of adiabatic parameter During the parameter change, =0.49MHz /µ s, =0.72µ s and µ =3.0rad are kept fixed. Dashed curve is the corresponding numerical result. Circles connected by dotted curve are the corresponding analytical results obtained by taking into account three pairs of branching points in Fig. 6.4. (Courtesy of D. Bouwmeester et al. )


117

minimum caused by the interference phenomena. In this way we can really observe local aspects of the gauge structure around the avoided level crossings.

6.3. Forces Induced by Born-Oppenheimer Approximation We have demonstrated that many diabolos and avoided level crossings appear in energy spectra of the classically-chaotic systems and that they induce the fictitious gauge potential in the adiabatic limit when external parameters are taken as frozen at each time. What phenomena are expected if the external parameters are slowly-varying variables like nuclear coordinates, rather than electric or magnetic fields? In studying the complex dynamics in microscopic cosmos, one encounters situations where a light (electronic) system with fast variables is coupled to a heavy (nuclear) system with slow variables. Since the dynamics of the whole system is quite complicated and hard to analyze, we separate fast from slow variables noting the difference in time scale of dynamical variables. Below we shall treat fast and slow degrees of freedom in terms of quantum and classical variables, respectively. An adiabatic approximation is first to solve the motion of fast variables for frozen values of slow variables and then to elucidate the slow dynamics under the averaged environment of the first motion. This approximation is called as the Born-Oppenheimer approximation (see also Messiah (1959)). The averaged energy of the fast system depends on the values of the slow variables so that its gradient w.r.t. slow variables provides a reaction force on the slow motion. We shall hereafter illustrate this point and then scrutinize the dynamics beyond the Born-Oppenheimer approximation to see chaos in quantum systems. Suppose that Hamiltonian for molecules with slow (nuclear) P, R and fast (electronic) p, r variables is (6.28) and apply the Born-Oppenheimer approximation to the complete eigenvalue problem For frozen R, the electronic system satisfies the eigenvalue problem

118

Chapter 6

for the adiabatic eigenvalue and eigenstate of with (R) and the fast (electronic) Hamiltonian, respectively . If one chooses an eigenfunction of the whole system (6.28) in a factorized form (6.29) (R) proves to satisfy (6.30a) unless

is degenerate. In (6.30a) (6.30b)

and (6.30c) are the fictitious gauge potential and effective potential, respectively. A(R) is essential in the vicinity of level crossings or conical intersections. Veff (R) consists of the adiabatic eigenvalue (R) and corrections due to the slow motion. From (6.30), we observe that the slow variables are subject to three reaction forces (Jackiw, 1988; Berry (ii) fictitious and Robbins, 1993): (i) Born-Oppenheimer magnetic (B= x A), (iii) fictitious electric ( in terms of the electrostatic potential The Lagrangian corresponding to the Born-Oppenheimer Hamiltonian in (6.30) is straightforward: L = MR 2 /2 + R • A Veff (R). (R implies time-derivative.) The kinetic energy in (6.30a) resembles that of a charged particle in the presence of a magnetic field. Consequently, upon quantization, the slow variables should satisfy the anomalous commutation relation (Jackiw , 1988; Kuratsuji and Iida, 1988) (6.31) where {Bk } denote the components of B. In the anomaly phenomena in modern quantum field theory, some symmetries of classical physics may disappear in corresponding quantum systems because of a procedure exploited in quantum-mechanical symmetry-violating


119

treatment. The relation (6.31) provides a typical example of this phenomenon. From a viewpoint of nonlinear dynamics the system described by (6.30) has a possibility of exhibiting chaos if the number of degrees of freedom is larger than that of the constants of motion. Will slow (and consequently fast) degrees of freedom be able to demonstrate chaos with the help of reaction forces in (6.30)? Our special interest lies in the genesis of chaos in the quantum (electronic) system with the nuclear system left classical. Below we shall pursue this theme.

6.4. Nonadiabaticity-Induced Chaos We shall show a significant consequence of the above discovery, by examining a concrete example exhibiting an isolated avoided crossing, i.e., a system of the Jahn-Teller type (Bulgac and Kusnezov, 1995). suppose the Hamiltonian Denoting a two-level electronic system by for a prototype of molecules with (nuclear) slow and classical P,Q and (electronic) fast and quantum variables as given by (6.32) where on the r.h.s. the 2nd and 3rd terms designate the potential for slow variables and the interaction between slow and fast variables, is a coupling parameter and f(Q) is a form factor. In respectively; terms of the density operator p, let the expectation value of be written as = r = (x, y, z). Then p can be expressed by (6.33) While p2=p in a pure state, the unequality p2
>1, (7.37) Secondly, the level density is available from the functional derivative, (7.38) (See Dyson (1962).) Using a large-order expansion of Legendre polynomials and after rescaling x 2x/L, the level density near x=0 is given by (7.39) for N>>1. The density (7.39) reaches a uniform value in the thermodynamic limit N, L with N/L a constant. In this limit, therefore, the Gibbs measure in (7.29) has proved to be successful in constructing the canonical ensemble. It should be emphasized that the present scheme has an advantage superior to Wigner's semicircle law for a Gaussian ensemble given by , which shows an unpleasant extensive behavior (~N½ ) in the thermodynamic limit. Finally, in terms of the probability (S) that an interval of size S is void of eigenvalues, the spacing distribution p(S) is defined by (7.40) The function (S) itself is reached (Mehta, 1967) by a limiting behavior of the probability given by (7.41) with u(x)=0, for gets

, and 1 for

Using (7.35) in (7.41), one

(7.42) where in terms of the S-dependent -1 spheroidal functions f0, f2 , f4,• • • . From (7.40) and (7.42), we find

Level Dynamics

147

which agrees with the universal level spacing distribution pOE(S) obtained for the Gaussian and circular orthogonal ensembles (Mehta, 1967). The well-known Wigner distribution, is merely an approximate distribution available from the "two-level model." The issue (7.43a) can be generalized to the cases v =2 and 4: For v=2, (7.43b) with quadratic level repulsion; for v=4 (7.43c) with quartic level repulsion. (7.43b) and (7.43c) are identical to pUE(S) for a Gaussian unitary ensemble and pSE(S) for a Gaussian symplectic ensemble, respectively. The statistical mechanics of gCM systems has thus explained why level spacing distribution for classically-chaotic systems (or spectra of quantum chaos) should obey the same universal distributions as those of random matrix theory. Curvature Distribution Although the random matrix theory conveys only statistical information when a set of nonintegrability parameters are fixed, the statistical mechanics of gCM system can also predict distribution functions for parameter-dependent quantities. Among such quantities, the most important one is level curvature (Fig. 7.5) given by (7.44) Curvature in (7.44) takes large values for a pair of adjacent levels at avoided crossings and constitutes a promising indicator of quantum chaos. In practice, numerical computation of curvatures Ki can be done by using the discrete form of the second-order derivative in (7.44) with a small but finite (In near-integrable regimes) anomalously large values for Ki may be available which would prevent us from

148

Chapter 7

constructing the curvature distribution. We shall therefore assume a fully-chaotic regime where the presence of universal curvature distribution can be expected.) In (7.44), is nothing but the acceleration of the i-th particle in gCM systems, so that (7.45) (see (7.23') ). Without loss of generality, we take x1 as a sample level and derive the probability density for curvature to take the value K: (7.46) with (7.47) where dMN,L is the Gibbs measure (7.29). From (7.23’), the equation K+ possesses N-1 roots {x1(k)} for any value of K. These zeros can be evaluated when K is large and positive, because is a function of x1 which diverges in the vicinity of levels different from x1 (see Fig. 7.6). Choosing K positive, we obtain (7.48) Consequently, one gets (7.49) with (7.50) Equation (7.49) is evaluated asymptotically for large positive values of K. After integration over the variables x1, {pn} and we obtain (7.51)

Level Dynamics

149

Fig. 7.6. Solutions of K+

where GN,L is given by the mean value over the joint probability density (7.30) as (7.52) In the thermodynamic limit and far from the hard walls, this expression is related to spacing density at zero spacing. Evaluation of the mean value (7.52) for a uniform spectrum of density p is quite easy. Let x0 be any eigenvalue in a sequence of ordered eigenvalues. Distribution in (7.52) is replaced by the function, for and =1 for Then (7.52) becomes a series of terms, each containing a pair of mean values: (7.53)

The m-th term is given by

150

Chapter 7 (7.54)

where pv(m)(S) is the the m-th order term in the spacing distribution. For m =0, pv(0) (S) is the nearest-neighbor spacing density given by with (7.43a)-(7.43c) for v =1,2 and 4 respectively. Since pv(m)(S) >v for m 1, (7.54) remains nonvanishing only for m =1 in the limit 0. In this limit, therefore, (7.53) becomes (7.55)

where pv(S)=pv(0)(S) and the constants Dv are /6, /3, and 24 /135, for v=1,2, and 4, respectively (see(7.43)). A similar result is available for K1) degrees of freedom (8.70) with =1,2,• • • ,M and nk=0,1,2,• • •. and mk represent mutuallyindependent closed paths and Maslov index, respectively (see Chap. 1); the second is the frequency rule (8.71) The former rule led Born and Jordan (1925) to the discovery of a commutation rule for canonically-conjugate observables, e.g. Eq. (8.67).

Towards Time-Discrete Quantum Dynamics

187

Two rules are embodied in the Heisenberg equation of motion for a Hermitian operator or matrix A: (8.72) The action J in (8.70) is calculable only when each classical orbit executes a closed path on the torus. In classically-chaotic systems, however, the collapse of tori prevents us from calculating the action on l.h.s. of (8.70) and consequently breaks the rule (8.70). The ultimate result should be that a logical foundation for the commutation rule (8.67) becomes ambiguous! It is a truth that the present-day quantum mechanics (Dirac, 1930) is constructed on the basis of two postulates (8.67) and (8.72) without any linkage to the old quantization rule in (8.69) and (8.70) and that its validity has been guaranteed by accumulation of experiments. But the foundation for the commutation rule (8.67) is traced back to (8.69) and (8.70). Although a modification of the commutation rule in chaotic systems might give rise to quantitatively small corrections in experiments, its role in the framework of quantum mechanics is beyond any conception. Nevertheless, the ambiguity about the commutation rule is not a serious problem in our construction of a time-discrete quantum mechanics, where both conditions and are employed. Let us discretize time as in units of a fundamental time interval ∆ t . By replacing the time differential of a Hermitian matrix A by (Euler's) time difference, (8.73) (8.72) becomes (8.74) This form inevitably breaks the unitary property and cannot keep the equal-commutation rule preserved in the time continuum limit. Therefore neither rule nor a way will exist by which to calculate the r.h.s. of (8.74), each time we should take a single time step forward. As an attempt to overcome the difficulty and to complete the timediscrete quantum dynamics, we propose: (i) to use forcibly the commutation rule (8.67) in calculation of the r.h.s. of (8.74) at each

188

Chapter 8

time step or (ii) to replace the r.h.s. of (8.74) simply by a Hermitian variant of the Poisson bracket as It should be noticed that, in some of systems including the example below, the two methods (i) and (ii) prove to yield the identical result. We shall now concentrate upon a spin system. The advantage of and the spin system is that its Hilbert space is finite-dimensional that, for any finite magnitude of quantum spin, the Hamiltonian matrix is also finite-dimensional with no necessity of artificial matrix truncation. As a concrete example, we choose a spin system driven by the x-polarized and time-periodic magnetic field B with period T= which is decribed by Hamiltonian (8.75) The 1st and 2nd terms on the r.h.s. of (8.75) represent an easy-plane anisotropy and Zeemann energy, respectively. (The -kick type driving field will not be considered, since our interest lies in discretization of time.) Let us first develop the classical treatment. Using Poisson brackets (8.76) and the Hamiltonian (8.75), the equation of motion for the spin S is given by

(8.77) The squared spin S2 is a constant of motion and is taken to be unity. Then, numerically iterating (8.77), we keep trajectory values at integer-multiples of period T. By introducing polar coordinates the Poincare' surface for a section in the plane is depicted in Fig. 8.7, which clearly exhibits a transition from tori to chaos with increasing B. To see a quantum analog of this transition, we shall analyze Heisenberg's equation of motion for the spin matrix, by keeping time continuous. The equation of motion in the quantum case reads


189

Fig. 8.7. Poincare' section for classical spin dynamics Polar coordinates are defined by (a) weak field case (A=1.0, µ B-0.3), (b) strong field case (A =1.0, µ B=1.0). The same values are used in Figs. 8.9 and 8.10.

(8.78) The quantum-mechanical counterpart of Lie algebra (8.76) is the commutation rule (8.79) From (8.78) and (8.79), we have (8.80a) (8.80b) (8.80c) Choosing as bases Fock states with Eqs. (8.80) might be regarded as three nonlinear high-dimensional matrix equations for Hermitian matrices. It is not legitimate, however, because (8.80) keeps the commutation rule (8.79).

190

Chapter 8

(The proof is given by induction.) In other words, all the matrix elements for (=0) are constants of motion, whose number equals to that of the (complex) variables, i.e., matrix a more rigorous elements of Sx, Sy and Sz. (In the case S= argument is required.) As a consequence, any solutions of (8.80) are not chaotic but recurrent. Following (8.73) and (8.74) together with the proposal below them, we shall now proceed to discretize the time variable as • (see Fig. 8.8) and rewrite the original Heisenberg equation of motion (8.80). The time differential on the l.h.s is taken as the Euler difference (8.81a) and expressions on the r.h.s. are replaced by using values at tj : (8.81b) (8.81c)

Fig. 8.8. Non-unitary time discretization: (a) time difference, (b) normalization of spin S.


191

As emphasized below (8.73) and (8.74), the time discretization like (8.81) will break the unitary property and cannot keep the equal-time commutation rule (8.79). Nevertheless, at each time step in the time evolution we shall have recourse to (8.79) to calculate the r.h.s. of (8.80). The resultant time-difference variant of (8.80) should take the following form: (8.82a)

(8.82b) (8.82c) Further, after each iteration of the difference equation (8.82), the S(j+1) will be made so that replacement (S(S+1)/Tr(S2(j+1) )) ½ S(j+1) the spin matrix should preserve the normalized magnitude S( S+1) (see Fig. 8.8(b)). Amazingly, (8.82) is also valid on regarding the classical equation (8.77) as the quantal equation for Hermitian matrices and on replacing the Poisson bracket on the r.h.s. of (8.77) by its Hermitian variant.

Fig. 8.9. Poincare' section of attractor for quantum spin dynamics. The spin coherent state with is used for the representation. Time difference t =T /3: (a) weak field case, (b) strong field case.

192

Chapter 8

Since, for S=1/2, (8.82) is a trivial linear map, we proceed to solve (8.82) numerically in the case S=1, where each component of S constitutes a 3 x 3 matrix. The result for valus of S at integer multiples of the period ) is given in Fig. 8.9. In this figure, the expectation value of S in a spin-coherent state is taken. The mapping (8.82) is not area-preserving and constituting a dissipative system, which is a weak point to be improved in future. However, both Fig. 8.9 and the autocorrelation function in Fig. 8.10(a) indicate the transition from regular to chaotic behaviors with increase of the magnetic field. An amusing feature can be found in the degree of breaking the commutation rule defined by (8.83) The time dependence of in Fig. 8.10(b) shows small and large fluctuations corresponding to regular and chaotic behaviors of the quantum spin, respectively. This result is sensitive to the magnitude of We find that, with decrease of the chaotic behavior is replaced by a regular one.

Fig. 8.10. (a) Autocorrelation function of <Sx(j)>. Dotted and solid lines are for weak and strong field cases, respectively. (b) Time dependence of Gray and black dots are for weak and strong field cases, respectively.


193

More precisely, there exists a threshold : Chaos emerges for > -1 depends on the degree of quantumness ( ~S ) and decreases as the classical limit S= is approached. The phase diagram in Fig. 8.11 depicts schematically a border curve between the collapse and genesis of genuine quantum chaos. The time-discrete variant of quantum mechanics thus describes a transition from regular to chaotic behaviors of quantum variables. In the present formalism, one can move from quantum to classical mechanics by taking both limits and Since the proposed map (transformation) (8.82) is not symplectic but dissipative, one is tempted to invent an improved map. For instance, (8.84) might be conceivable as a spin analog of the Moncrief’s timediscretization scheme in (8.62) or (8.631, in terms of a unitary matrix given by

(8.85)

Fig. 8.11. Phase diagram for occurrence of quantum chaos.

194

Chapter 8

Equation (8.85) is an approximation to the genuine unitary matrix In fact, Although (8.84) with (8.85) is a quantum version of the symplectic map for the classical spin dynamics, it preserves the commutation rule (8.79). Despite exhibiting a feature of the nonlinear high-dimensional map, (8.84) with (8.85) is a linear map yielding absolutely regular orbits. Fluctuations of Fundamental Time Step The way of time discretization should be justified by means of time quantization. Meanwhile, there exists no good idea of such justification. While the fundamental time step ∆ t has so far been assumed constant, let us replace it by the stochastic variable (with the mean and the variance obeying the uniform distribution defined in the region In this case, the erratic behavior of the quantum spin is anticipated without resorting to the non-unitary transformation, which will also be described below. We reconsider the ordinary unitary transformation (8.84) for the S=1 quantum spin under a unitary operator

(8.86) where . When is small enough, there is no difference between the time-continuous and time-discrete quantum spin dynamics, and the quantum dynamics exhibits merely the quantum recurrence, irrespective of the magnitude of the driving field B. If At will become of the order of however, the quantum spin dynamics shows a variety of features, depending on B. Figure 8.12 is the temporal behaviours of the expectation value Sx(j) in the same spin coherent state as used in Fig. 8.9. Although the dynamics with a fluctuating step ∆ ti is identical to the one with the constant step A t in the weak field case, the discrepancy between two kinds of dynamics is serious in the strong field case. Consequently, despite the unitary property of the quantum dynamics with a fluctuating step, the theorem of the quantum recurrence is broken for the strong field case. In fact, the time series of <Sx(j) > at the successivec discrete times (~ integer multiples of the period T) exhibit a transition from periodic to erratic behaviours with increasing the driving field.


195

Fig. 8.12. Time series of <Sx(j)> for the case of fluctuating and time difference (a) weak field case (A=1.0, µ B=0.05), (b) strong field case (A =1.0, µ B=1.0) The same values are used in Fig. 8.13. Broken lines are for the case with no fluctuations, ie., for any i.

Fig. 8.13. Autocorrelation function of <Sx(j)> for the case of fluctuating (a) weak field case, (b) strong field case.

:

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Figure 8.13 is the autocorrelation function of <Sx(j)>. We see the rapid decay and persistence of the correlation for the strong and weak field cases, respectively. Therefore, the fluctuations of uncover the difference between chaotic and regular dynamics that is hidden in both the time-continuous and discrete-time unitary quantum dynamics. the apparently chaotic behavior discussed here is Nonetheless, caused by the stochastic behavior of the fundamental time step and not by the deterministic mechanism. It is not valid to call the present erratic behaviors as the deterministic chaos. We are therefore led to a conclusion that, unless we should break the unitarity in quantum dynamics, any kind of time-difference equation could fail to yield a chaotic solution. To obtain genuine chaos in quantum mechanics, it is crucial to pursue a non-unitary time evolution.

8.8. Problems to be Examined In this chapter, after a thoroughly comprehensive investigation into the role of time discretization in the (originally) time-continuous classical dynamics, we have analyzed various time-discrete variants of Heisenberg's matrix equation of motion. To suppress an artificial nonlinearity arising from finite truncation of infinite-dimensional matrices, we have concentrated on the spin system. Any kind of time-difference equation preserving unitarity proves to exhibit no indication of chaos, because of its reduction to the strictly linear map for the wavefunction. We have therefore proposed a non-unitary time evolution of spin matrices, displaying a numerical evidence of transition from tori to chaos, i.e., a genuine quantum chaos. Quantum chaos is shown to appear when the fundamental time step is larger than some threshold below which it dissappears. itself has a maximum at the quantum limit S=l/2 and decreases as the classical limit S= is approached. The scheme proposed above, reducing exactly to the time-continuous Heisenberg equation of motion is compatible with the existing framework of in the limit quantum mechanics. As for a candidate for , one may point out, for instance, the time interval for a photon to traverse the electron diameter; actually it is meaningless to imagine the continuous time variation within such an unmeasurable interval. It is desirable to carry out the timedependent measurement with high precision for the purpose of


197

observing quantum chaos induced by the time discretization, however difficult a task that would be. To place the present scheme on a more sound basis, there are many serious queries to be addressed: (1) We have concentrated on a special kind of non-unitary evolution. Which one is the most substantial among various kind of non-unitary time evolutions? (2) We have investigated the Heisenberg equation of motion, which is convenient for seeing relationship with the classical equation of motion. But, what will be a corresponding discrete-time evolution of wavefunctions? Is it still possible to provide a probabilistic interpretation of in the case when the unitarity is not satisfied any more? (3) Can we extend the present scheme to more general dynamical systems with variables p , q obeying ordinary canonical commutation rules ? (4) Is it possible to improve the present trial by inventing a time-discrete symplectic but still non-unitary quantum dynamics ? All these questions remain to be answered.

References Amit, C., Ching, E. S. C., Kadanoff, L. P., and Rom-Kedar, V. (1992). J. Nonlinear Sci. 2, 9. Bender, C. M., et al. (1985). Phys. Rev. Lett. 55, 901. Born, M., and Jordan, P. (1925). Z. Phys. 34, 858. Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. Oxford: Clarendon Press. Gelfreich, V. G., Lazutkin, V. F., and Svanidze, N. V. (1994). Physica D71, 82. Hakkim, V., and Mallick, K. (1993). Nonlinearity 6, 57. Kruskal, M. D., and Segur, H. (1991). Stud. Appl. Math. 85, 129. Lazutkin, V. F., Schachmannski, I. G., and Tabanov, M. B. (1988). Physica D40, 235. Lee, T. D. (1987). J. Stat. Phys. 46, 843. Moncrief, V. (1983). Phys. Rev. D28, 2485. Nakamura, K. (1993). Quantum Chaos: A New Paradigm of Nonlinear Dynamics. Cambridge: Cambridge University Press. Nakamura, K., and Hamada, M. (1996). J. Phys. A29, 7315. Poincare', H. (1890). Acta Math. 13, 1. Schrödinger, E. (1926). Ann. Phys. (Leipzig) 81, 109. Tovbis, A. (1994). Commun. Math. Phys. 163, 245. Tovbis, A., Tsuchiya, M., and Jaffe, C. (1996). Preprint. Weinberg, S. (1989). Ann. Phys. (N.Y.) 194, 336.

Chapter 9 Conclusions and Prospects As we have hitherto studied, the present framework of quantum mechanics cannot yield genuine chaos characterized by positive Lyapunov exponents. The quantum dynamics of systems with a few degrees of freedom does not exhibit temporal behaviors other than periodic or quasi-periodic oscillations characterized by discrete energy spectra. It is true that quantum dynamics of many-body open systems may have nonzero generalized KS entropy extended to systems obeying the non-commutative algebra, but such dynamics cannot be endowed with any positive Lyapunov exponent. So far much space has been devoted to the semiclassical quantization of chaos and random matrix theory, which are exploited by a growing number of researchers in the analyses of chaos and quantum transport in mesoscopic cosmos. In Chap. 8, we have even searched for a way to reconcile quantum with chaos . Let us now summarize these results and address future prospects. Chaos and Quantum Transport in Mesoscopic Cosmos We have developed both classical and quantum-mechanical investigations on open concave and convex billiards whose realization can be found in semiconductor microstructures. The conductance g(B) as a function of the magnetic field B in open circle (Cl) and stadium (Sd) billiards (quantum dots) has proved to be related to the feature of classical dynamics. Fluctuations of g(B) are shown to be dependent largely on the stability of phase space in the underlying classical dynamics of closed billiards. In the case of Cl billiard, the regular modulation of periodic orbits in the phase-space structure gives rise 198

Conclusions and Prospects

199

to regular oscillations of g(B). On the other hand, the global chaos and genesis of successive tori with increasing B in the case of Sd billiard are responsible for slow and rapid variations of g(B), respectively. The theoretical result is qualitatively consistent with Marcus et al.'s experiment (1992). Furthermore, the classical conductance, calculated on the basis of the bouncing Larmor-orbit picture, has been shown to reproduce most of the locations of peaks in the coarse-grained version of g(B). Real nanoscale structures are accompanied by extrinsic randomness, e.g., corrugation of hard walls, impurities and thermal noises. The rapid progress of advanced technology will smear out these obstacles that prevent us from comparing theory with experiment. The quantum theory of chaos is thus entering an era that will see its experimental verification by means of quantum transport in mesoscopic devices. Currently, theoretical interests are focused on a simple theme: (1) showing universality of conductance fluctuations on the basis of random matrix theory, (2) deriving the S matrices directly by extending Gutzwiller's semiclassical trace formula to open systems. Nevertheless, the observed magneto-conductance of Marcus et al. would demand a much deeper insight: The discrepancy of the transition point by more than order of magnitude between the theory and experiment (see Chap. 4) is very serious, and one should accept a challenge to solve this puzzle which, meanwhile, can be solved by neither the semiclassical nor the quantum theory. For the purpose of enriching the above assertion, a comparative study on the Lyapunov exponents and magneto-conductance G(B) in open square and single Sinai billiards has also been made. It has been discovered that conductance fluctuations depend again on the stability of mixed phase space in the underlying classical dynamics. In the Sinai billiard case the classical phase space is globally stable against B, while for the square billiard it is globally unstable, so long as a low B-field region is concerned. Smoothed (coarse-grained) conductance Gcg (B) reveals a continuous transition between chaos and tori. In the case of a square billiard, the correlation field Bc of the smoothed conductance is shown to vary irregularly with respect to B, while in the case of a Sinai billiard it decreases monotonically with increasing B. More careful study revealed that two kinds of correlation fields Bc s are relevant: Bc of Gcg (B) (low frequency component) and that of G(B) - Gcg(B) (high frequency component) have turned out to mimic the average Lyapunov exponent and scaled standard

200

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deviation of Lyapunov exponent , respectively. The fluctuation of Bc in the case of square billiard is attributed to the ghost orbits with ill-converged large or small Lyapunov exponents proper to a scattering (not periodic) orbit. In the Sinai billiard case, the geometry of the billiard forms an Aharonov-Bohm ring, so that A-B oscillation is observed in a weak field region. While the A-B effect suppresses symptoms of chaos, both Bc s for Gcg(B) and G(B) - Gcg(B) are nicely related to Lyapunov exponents and therefore capture the quantumclassical correspondence. A pioneering experiment by Weiss et al. (1993) on quantum transport in mesoscopic Sinai billiards (i.e., antidot arrays) was also sketched. Despite the elegant interpretation of some of spectral properties, there remains an open question concerning the underlying low-B field anomaly, which should be interpreted on the basis of isolated unstable periodic orbits. In particular, it is possible to gain alternative insight into the fluctuations in the vicinity of zero field when the system is fully chaotic and no stable periodic orbit survives. Semiclassical Quantization of Chaos The semiclassical theory of chaotic scattering is a powerful tool to describe the transport properties in the zero-field regime. Its application to convex hard disk systems (whose realization can be found in finite antidot arrays at the interface of semiconductor heterojunctions) have the following advantage: (1) The systematic enumeration of all periodic orbits is possible with the help of symbolic dynamics. (2) So long as one is concerned with the case of a large degree of opening, only short periodic orbits contribute substantially to the trace formula. As a consequence, the conditional convergence of the trace formula is possible by resorting to the Ruelle zeta function, eliminating the problem of the divergence due to exponential proliferations of periodic orbits (bouncing between disks). The locations of poles (i.e., scattering resonances) of the semiclassical S matrix are in good agreement with those evaluated by the exact quantum-mechanical theory. The region void of resonances in the complex plane is elucidated. The semiclassical theory of chaotic scattering has not only a conceptual significance for uncovering the quantum-classical correspondence, but also an advantage to be more practical than the exact quantum theory which, in applications, will be confronted with computational limitations in both c.p.u. time and memory area. As described in Chap. 2, the autocorrelation function of the trace


201

formula and that of S matrices can be calculated approximately, giving insight into the transport phenomena in the mesoscopic cosmos. The semiclassical scattering theory also predicts the zero-field Lorentzian peak and the Al'tshuler-Aronov-Spivak effect in the wave-number averaged reflection probability for ballistic chaotic billiards (see Chap. 4). A serious problem, however, arises from the semiclassical scattering theory applied to quantum transport in actual mesoscopic microstructures (quantum dots), where conducting lead wires are connected to cavities. There the semiclassical quantization of chaos will be incomplete unless anomalous orbits due to wave diffraction are incorporated besides the scattering orbits. This notion holds for a series of interpretations since Jalabert et al. (1990) of the conductance fluctuations in the ballistic quantum transport. It is very difficult to fully describe the effect of diffraction in terms of the particle picture. (Here we have in mind the case where the trace formula is combined with the Kubo formula.) At the same time it should also be emphasized that the semiclassical theory of chaos or Gutzwiller's trace formula would not be the ultimate theory on "quantization of chaos." The semiclassical theory is based on the assumption that the Schrödinger-Feynman framework of quantum mechanics should be effective even in systems exhibiting (classical) chaos. It is therefore not so surprizing to find a good agreement between poles of semiclassical zeta functions and those of quantum-mechanical S matrices in hyperbolic billiards without any bifurcation. So long as one stays within the framework of the Schrödinger-Feynman quantum mechanics, the trace formula indeed provides the most valuable tool for exploring many interesting topics lying on the borderline between quantum and classical mechanics of chaotic systems. However, the calculation of the trace formula applied to fully-chaotic bounded systems will encounter a serious problem of nonconvergence in the series sum due to the exponential proliferation of periodic orbits. This problem may be partly overcome either by smoothing the density of states or by inventing a way to achieve conditional convergence by means of the Ruelle zeta function. For bounded systems, however, the eigenvalues computed from the trace formula are not real! To resolve this problem, one should improve the trace formula so as to include higher-order terms in which will demand more complicated mathematics. Further, in generic and mixed systems with elliptic islands (KAM tori) coexisting with a chaotic sea (e.g., billiards in the magnetic field), even

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the symbolic coding of periodic orbits is much less obvious. Since the fundamental law should be as simple as possible, we are here tempted to improve the Shrödinger-Feynman formalism of quantum mechanics for classically chaotic systems in order to capture a much simpler classical-quantal correspondence in the semiclassical limit. The pursuit of this kind should be guided by, and connected with, a growing number of experiments on nanoscale structures in the mesoscopic cosmos. Random Matrix Theories Despite an accumulation of works based on the identification of energy spectra in classically-chaotic systems and random matrix theory, there are many counter-examples: those systems possessing GOE level statistics cannot always exhibit chaos in the corresponding classical dynamics. The quantum theory of chaos has a much richer content than the random matrix theory. We have recognized in Chap. 7 why the quantum spectra of classically-chaotic systems should obey the same universal level statistics as in random matrix theory. There exist universal classical dynamical systems, i.e., the generalized Calogero-Moser (gCM) and Calogero-Sutherland (gCS) systems, lying behind quantum systems which are in general mixed in the corresponding classical dynamics. The statistical mechanics of gCM (or gCS) systems is very fruitful, leading to the curvature distribution as well as the major results of the random matrix theory. It is more general than the framework of random matrix theory. As explained in Chap. 7, the statistical mechanics of gCS systems under some constraints will be able to provide the level statistics in the intermediate and mixd regime. I have a strong criticism against the prevailing tendency to reduce the nature of chaos in quantum systems to that of random matrix theory. While many current researches concern the quantum irregular spectra mediated by fully-chaotic systems, most of the classical dynamical systems possess mixed features implying that KAM tori coexist with chaos. It is desirable to derive an intermediate statistical behavior linking the Poisson and Gaussian ensemble statistics from first principles, without being satisfied with Brody's empirical formula. There would however be no universal statistical behavior in intermediate regions, which is highly system-specific. Choosing the number variance that describes the long-range (rather than short-range) correlation of ensemble of levels, we have pointed out one interesting


203

channel introduced by Gaudin as early as 1966. D y n a m i c s b e y o n d B o r n-O p p e n h e i m e r A p p r o x i m a t i o n In an attempt to construct chaotic dynamics in quantum systems within the present formalism of quantum mechanics, we have analyzed in Chap. 6 the dynamics of systems with a few degrees of freedom beyond the Born-Oppenheimer approximation. In the adiabatic limit, when time scales in the dynamics of molecular complex systems are radically different between slow (nuclear) and fast (electronic) degrees of freedom, the reaction forces (due to fictitious magnetic and electric fields) are indeed exerted on the slow (nuclear) degrees of freedom that are treated classically. But the quantum (electronic) subsystem cannot exhibit chaos. In the dynamics beyond the adiabatic approximation, however, we can see a genesis of chaos in the quantum system or the genuine quantum chaos in both bounded and open systems. We saw a nice example illustrated by Bulgac and Kusnezov (1995). From a viewpoint of nonlinear (classical) dynamics, most of molecular kinetics in the microscopic cosmos exhibit a possibility to generate chaos. However, if each molecular complex system were quantized at the outset, this possibility would be smeared out. Eventually, it is difficult to see chaos in quantum systems without artificial approximations. This means that the present formalism of quantum dynamics should be augmented so as to accommodate temporal chaos. Towards a Challenge to Reconcile Quantum with Chaos For systems showing chaos in the underlying classical dynamics, we have proposed an alternative formalism of quantum dynamics which exhibits the results of quantum chaos while recovering the existing Heisenberg or Schrödinger equation in a suitable limit. The progress in nonlinear dynamics and chaos over past decades has revealed that chaos is easily generated from the time-difference variant of the time-continuos integrable system. Motivated by a great significance of time-discretization of the originally timecontinuous classical dynamics and also by our expectation to understand more clearly the uncertainty principle between time and energy in our microscopic cosmos, we have chosen a way to discretize the time t in quantum dynamics. In this context, we have scrutinized a time-discrete variant of Heisenberg's equation of motion that has a direct correpondence with the classical equation of motion. The new

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framework of quantum mechanics assumes both and 0 and can be reduced to classical mechanics as well as the present formalism of quantum mechanics in suitable limits. To be explicit, we propose a non-unitary time evolution of spin matrices, displaying a numerical evidence of transition from tori to chaos, i.e., genuine chaos characterized by a positive Lyapunov exponent. We are faced with the demand to proceed to improve the present trial by inventing a timediscrete symplectic but still non-unitary quantum dynamics. The time discretization of Heisenberg equation of motion is not a unique way to generate quantum chaos. Other attempts may be conceivable to pursue the possibility of quantum chaos. Among them, we shall mention briefly (1) the extension of Liouville's equation for the density operator, i.e., a generalization of Schrödinger's equation, and (2) the method of using a time-continuous mesurement. (1) Attempt by Prigogine's School. By resorting to the statistical description, Prigogine and coworkers (Petrosky and Prigogine, 1994) are trying to incorporate chaos into the quantum dynamics. They choose to describe the evolution of an ensemble of states by means of the Liouville operator acting on the density operator: (9.1a)

with (9.1b) This Liouville equation is identical to Shrödinger's equation for and the integrable case. In fact, we have

in

for pure and mixed states, respectively. By attributing the effect of chaos to Poincare' resonance, i.e., the resonance between different degrees of freedom, Prigogine et al. assert that in the presence of Poincare' resonances the nonunitary diffusion process is combined with the reversible process. In this anomalous process the eigenvalue problem for the Liouville operator (9.1b) has a solution outside the Hilbert space: The solution involves a singular term breaking the time reversal symmetry. Therefore the density operator cannot be reduced to wavefunctions any more. The elaborate attempt by Prigogine and coworkers, however, is directed towards deriving the irreversibility (or determining a direction for the arrow of time) with a help of chaos and thus is not

Conclusions and Prospects concerned with generating the positive Lyapunov exponent.

205 quantum chaos

characterized by

a

(2) Quantum Chaos in Continuous Measurements. Bearing a photon-counting experiment in mind, we are presently investigating the possibility to envisage quantum chaos in continuous measurement. The measurement via photon-counting was originally proposed to realize the Schrödinger's cat state in the laboratory (Ueda et al., 1990). Let the whole system be composed of the measured system (i.e., the subsystem to be measured) and the detector (i.e., measurement apparatus). The measured system, consisting of a twolevel atom and the near-resonant photon field, is confined to a cavity. This measured system, called the Jaynes-Cummings model, is known to show chaos if the photon field could be treated classically without resorting to the rotating-wave approximation. On switching the photon detector on at t=0, one will continue to measure photons emitted through a small hole of the cavity. The continuos read-out of the measurement information will give a continuous reaction on the subsystem and thereby lead to a non-unitary evolution of the density operator ρ of the measured system. The measurement is of either the demolishing or non-demolishing type, depending on whether each detection of a photon is possible with or without an annihilation of the photon inside the measured system. The longer the measurement is continued, the more the measured system would be forced to couple with the large degrees of freedom outside, dissipating the information and losing the quantum coherence. After all, the measured system is expected to possess a feature of the classical system which is able to show a chaotic response. In the measurement process, however, results of measurement are provided in an indeterministic way, so that one cannot predict the observed value in advance. Therefore the quantum chaos in the continuous measurement inevitably involves stochasticity. Investigations similar to ours is being made by Mensky (1995). Among others, however, the idea developed in Chap. 8 to construct an alternative framework of quantum mechanics by timediscretization would provide the most promising way to unify quantum and chaos. Epilogue As we have insisted throughout the book, the genesis of chaos is disturbing the quantization condition of the adiabatic invariants

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(Einstein, 1917), which is a logical foundation of modern quantum mechanics. This condition is, on the one hand, a basis for the commutation rule in the Heisenberg formalism of quantum mechanics and, on the other hand, is identical to Schrödinger equation for the wavefunction that is dual to the particle executing periodic or quasiperiodic motions. The latter statement is not changed by the probabilistic interpretation of the wavefunction. Once the framework of quantum mechanics was established, however, all questions about its very logical foundation seem to be forgotten, by our paying major attention to the application of the theory to practical results, e.g., superconductivity and quantum Hall effect. It is amazing that these essential questions remain still unanswered, despite the accumulation of fruitful and practical issues of quantum mechanics. In this book we have explained the semiclassical theory of chaos or Gutzwiller's trace formula (Gutzwiller, 1990). While still many activities are concerned with its improvement, e.g., inclusion of corrections in etc, the trace diffraction effects, higher-order formula is not the ultimate theory on quantization of chaos. This formula is based on the assumption that Schrödinger-Feynman's formalism of quantum mechanics should be effective even for classicallychaotic systems. The genesis of chaos, however, makes the above asumption groundless and therefore demands that we enrich or revolutionize the present framework of quantum mechanics (Nakamura, 1993). Our considerable efforts have also been devoted to describing the quantum-classical correspondence ( quantum symptoms of chaos) within the present formalism of quantum mechanics. While the description has been largely concerned with quantum transport and irregular energy spectra, wavefunction features are much less trivial. In particular, many works have elucidated (i) the suppression of chaotic diffusion in the wavepacket dynamics and (ii) the scars of unstable periodic orbits embedded in wavefunctions (Giannoni et al., 1991). However, the major interest throughout the book has centered on the pursuit of a quantum mechanism which generates chaos. So far there is no experimental report to suggest a breakdown of quantum mechanics when applied to classically-chaotic systems. This might be due to the presence of competition between fluctuations caused by determinisic chaos and those by thermal noise and random potentials. Nevertheless, there is a strong possibility to envisage a limitation of quantum mechanics which cannot accommodate chaos, owing to a


207

rapidly-developing high technology like (1) ultra-low-temperature technology, (2) ultra-small scale fabrication of highly-purified quantum dots and antidots, and (3) measurement on ultra-short time scales. As candidates capable of generating chaos in quantum mechanics, we mentioned: (1) the time-discretization of Heisenberg's equation of motion, (2) a generalization of Liouville's equation so as to include dissipative process, and (3) continuous measurement, e.g., via photon counting. Other kinds of new quantum formalisms would also be conceivable which should reduce to the Hamilton-Jacobi equation in the classical limit It is crucial that theoretical efforts in this direction keep stride with experimental verifications via advanced high technologies such as quantum transport in nanoscale quantum dots or antidots of high purity and with suppressed noise (Beenakkar and Houten, 1991; Akkermans et al., 1995; Nakamura, 1997). To conclude, the pursuit of quantum chaos to reconcile quantum with chaos deserves an extremely great effort, comparable to the theoretical and experimental activities around the blackbody radiation of one century ago that led to the discovery of quantum mechanics.

References Akkermans, E., Montambaux, G., Pichard, J.-L., and Zinn-Justin, J., eds. (1995). Mesoscopic Quantum Physics, Proceedings of Les Houches Summer School. Amsterdam: North Holland. Beenakker, C. W., and van Houten, H. (1991). In Solid State Physics: Advances in Research and Applications, H. Ehrenreich and D. Turnbull, eds. New York: Academic. Bulgac, A., and Kusnezov, D. (1995). Chaos, Solitons and Fractals 5, 1051. Einstein, A. (1917). Verh. Dtsch. Phys. Ges. 19, 82. Gaudin, M. (1966). Nucl. Phys. 85, 545. Giannoni, M. J., Voros, A., and Zinn-Justin, J., eds. (1991). Chaos and Quantum Physics, Proceedings of the NATO ASI Les Houches Summer School. Amsterdam: North-Holland. Gutzwiller, M. C. (1990). Chaos in Classical and Quantum Mechanics. Berlin: Springer. Jalabert, R. A., Baranger, H. U., and Stone, A. D. (1990). Phys. Rev. Lett. 65, 2442. Marcus, C. M., et al. (1992). Phys. Rev. Lett. 69, 506. Mensky, M. (1995). Chaos, Solitons and Fractals 5, 1381.

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Nakamura, K. (1993). Quantum Chaos : A New Paradigm of Nonlinear Dynamics. Cambridge: Cambridge University Press. Nakamura, K., ed. (1997). Chaos and Quantum Transport in Mesoscopic Cosmos, Special issue of Chaos, Solitons and Fractals 7. In press. Petrosky, T., and Prigogine, I. (1994). Chaos, Solitons and Fractals 4, 311. Ueda, M., Imoto, N., and Ogawa, T. (1990). Phys. Rev. A41 , 3891. Weiss, D., et al. (1993). Phys. Rev. Lett. 70, 4118.

Index Adiabatic approximation 117,122,123 change 104 energies 109 invariant 12, 13,16 limit 104 potential surfaces 120 state 108 transport 105 Aharonov-Bohm (AB) effect 22, 200 oscillation 48, 71, 76, 200 Allan variance 38 Al'tshuler-Aronov-Spivak effect 75 Anomalous commutation relation 118 Anomalous fluctuations 59 Anomaly phenomena 118 Antidot 70, 80, 100 arrays97, 98, 200 Aperiodic fluctuation 66 Area-preserving 5, 52, 192 Asymptotic expansions beyond all orders 162,163, 175, 180, 181 Autocorrelation function 42, 60, 71, 200 Avoided (level) crossing 102, 104, 155,156,159 multiple 127 Ballistic conductance fluctuation (BCF) 49, 50 transport 201 Berry phase 154 Billiard circle (Cl) 46-47, 59-63, 74, 75, 196 concave 46, 54 convex 46, 79, 80 four-disk 79, 96 hyperbolic 80,201

Sinai 23, 46, 66, 70, 75, 80, 98, 199 square 67, 71, 73, 75,199 stadium (Sd) 13, 46, 47, 59-63, 74, 75, 198 Birefringence 114 Birkhoff coordinates 5 1 Blackbody cavity 12 Blackbody radiation 12, 15,207 Bloch bands 35 Bohr magneton 108 radius 41 Bohr-Sommerfeld’s quantization 11, 15 Borel summability 163 summable 170, 171 summation 170 transformation 170,173 Born-Oppenheimer approximation 106, 117, 121 dynamics beyond 203 force 118, 121 Brody's empirical formula 202 Brownian motion 9, 128, 129, 130 fractional 29 Bunimovich-Sinai curvature formula 91 Canonical ensemble 143 intermediate 143 Canonical transformation 2,3 Cantori 26 Circular orthogonal ensembles (COE) 147 unitary ensembles (CUE) 151-154 Classical conductance 61 Classical distribution function 9-11 Classical-quantum correspondence 11, 181 Classical transition probability 25

209

210 Classical transmission probability 28 Coarse-grained conductance 71 Codimensions 155 Coherent state 9 Comb-like structure 67 Complex crossing point 109,110 Complex Grassmannian sigma model 159 Complex time plane 168 Conditional convergence 21, 89, 200 Conductance fluctuation 26, 73, 201 ballistic (BCF) 49, 50 universality of 199 Conservative systems 7 Coulomb blockade 63-65 Cross-over time 11 Crossroads 13, 80, 96 Cumulative distribution function 93, 94 Curvature distribution 147-150 Cyclotron radius 47 De Broglie particle 182 relation 13 wavelength 92 wave-particle dualism 11 Degeneracies of eigenvalues 154 Degenerate perturbation theory 103 Degree of randomization 9 Demkov-Osherov Hamiltonian 137 Density of states 24, 85 Destructive phase interference 26 Diabatic representation 157 state 110 Diabolos 156 Diamagnetic Rydberg atoms 136,139 Diffusion coefficient 82 Dirac monople 12 1 Discretized orbits 8 Double-barrier structure 31, 32, 39, 43 Double-cone 156 Double-well potential 163,166 Driven nonautonomous systems 130 Duffing equation 164 Dwelling time 67 Einstein-Brillouin-Keller (EBK)'s quantization rule 19 Energy resonator 12 Ensemble

Index Laguerre 144 Legendre 145 symplectic 145 Equal-time commutation rule (ETCR) 185, 191 Eriscon's fluctuation 26 Essential singularity 180 Evanescent waves 56 Exponential proliferation 21, 25, 57, 80 of periodic orbits 201 Exterior differentials 105 Fermi energy 55, 98 level 23 velocity 96 wave length 66 Feynman's path- integral 16 1/f fluctuation 35 Fictitious gauge field 155 potential 104, 106,107, 109, 118 Field-theoretical model 155,158 Filamentary repeller 83 Fixed points elliptic 5 hyperbolic (HFP) 5, 20, 163, 177, 178 Flicker floor 38 Flux quanta 23 Fokker-Planck equation 129 Fractal conductance fluctuations 26 Fredholm determinant of an integral kernel 144 Fugacity 154 Fundamental time step 194 Gauge potential 111, 113, 158 Gauge structure 102 global 108 local 108 Gauge transformation 157 Gaussian orthogonal ensemble (GOE) 46,128, 130, 136,139,141, 202 smoothing 10 symplectic ensemble 147 unitary ensemble 147 wave packet 10 white noise 130 Generalized Calogero-Moser (gCM) system (model)

Index 134-136, 148, 151, 157, 158, 202 infinite 142 statistical mechanics of 142 Generalized Calogero-Sutherland (gCS) system 135, 151, 202 Genuine quantum chaos 32,196 Geometric phase 104-106,114 Ghost orbits 67, 69, 70, 76, 200 Gibbs measure 143,146,148,152 Green function 16, 56, 58, 70, 84 theorem 56, 84 Hamilton's principle 17 Hard disks 80 Hartree approximation 43 equation 40 Heisenberg equation of motion 182, 183,190, 196 equation of motion for spin matrix 188 matrix mechanics 11 uncertainty principle 2, 10 Hellmann and Feynman's theorem 132 Henon maps 181 Heteroclinic structure 163 Heterostructures 39 Holonomy 106 Homoclinic orbits 20 point 5,7 structure 7,175, 181 Hyper-energy surface 155, 156 Ill-converged 70 Impurity potentials 14 Inelastic mean free paths 66 Information dimensions 92 Intermediate distribution 139 Intermittent chaos 36, 38 Internal equation 168 Intersection angle 181 Irreducible closed contour 18 Jaynes-Cummings model 205 Joint distribution function 151,152 Joint probability density 130,144, 149 KAM tori 5, 9, 26, 46, 53, 60, 67, 136, 151,202

211 collapse of 4 Kicked rotator 131 Kolmogomv-Sinai(KS) entropy 1, 7, 9, 21, 32, 42, 65, 81, 181, 198 Krönig-Penny model 31 K system 46 Lagrange multiplier 152 Laminar oscillations 38 Landauer formula 70 Landau gauge 55 Landau level 98 Landau-Zener(LZ) model 108 Gaussian twisted 115,116 winding 110,111 Landau-Zener(LZ) transition 111,124, 137 Langevin equation 130 Larmor-orbit bouncing 62, 63, 199 Larmor radius 50, 68, 83 Law of large number 38 Lead wires 70 Leapfrog method 183 Lebesgue measure 81 of periodic orbits 20 Legendre orthogonal ensembles 130 Level degeneracy 102 Level-dynamics 128 Level spacing 64 distributions 128 Liouville equation 10 extension of 204 Lyapunov exponent 7, 20, 32, 35, 37, 42, 52, 55, 65, 67-69, 71, 75, 81, 82, 87, 90, 96, 181, 198, 199 Magneto-conductance 47, 49, 75, 199 Manifolds stable 7, 170, 179 unstable 5, 7, 162, 166, 170, 179 Many-body effect 43, 47, 65 Markovian process 38 Maslov index 13, 17, 18, 25, 28, 58, 186 Maxwell-Boltzmann distribution 137 Maze-like structure 10 Mean free path 98 Mean level spacing 24 Mesoscopic devices 13 structures 47

212

Index

Mexican-hat potential 121,122 Mixed phase 27 phase space 50, 75, 199 systems 28 Moncrief’s time-discretization 193 Monodromy matrix 20, 52, 87, 90 Monopoles 159

recurrence theorem 11 resonance 204 surface of section 5, 90, 188 Poisson brackets 134 Poisson distribution 137,139 Probability density function 9 Pseudo-chaos 31-32, 35, 39 Pseudo-time 132

Nanoscale structures 13 Nikitin's model 113 Noises fractional 38 thermal 14, 63, 199 NonAbelian gauge field 157 Nonadiabatic transitions 107,108,122 transport 108, 109 Non-Gaussian distribution 64 Nonlinear Hartree-like equation 35 Non-unitary diffusion process 204 quantum dynamics 197,204 time evolution 196 transformation 194 Number variance 151-153

Quantization of action 186 of chaos 15, 19, 201 of integralsystems 17 Quantum dot 76,201 Quantum recurrence 11 Quantum transport 198, 199 ballistic 201 in superlattice 32 Quantum wires 67 Quasi-time 128,131 Quaternionic Hamiltonians 103 Quaternions 135

Parallel transport of quantum eigenstates 106 Pauli matrices 106,135 Peierls phase factor 70 Periodically-pulsed system 131 Periodic orbits exponential proliferation of 30 isolated prime 87 primitive 20 scars 46 stable 20 symbolic coding of 22, 29 unstable 7, 19, 82 Perron-Frobenius operator 81 Persistent currents 22. 23 Phase coherence length 98 Phase droplet 7 Phase-integral method 110 Phase liquid 10-11 Phase space 71 Photon-counting experiment 205 Poincare' linearized map 20, 59, 67, 87 map 5, 50-51, 58, 90

Random forces 129 Random matrix theory 63, 94-96, 136, 142, 144, 147, 199-202 Reaction forces 118 Recurrent phenomenon 2 Reduced action 20, 25 Resistance anomalies 98 Riemann-Siege1 type resurgence 21 Riemann's zeta function 21 Ruelle zeta function 21, 30, 89, 95, 200 Saddle-point approximation 19 Scattering orbits 25 resonances 82, 87, 92, 93 Schrödinger equation 55, 96, 108, 120, 181,206 descrete variant of 185 nonlinear 43,164 time-dependent 40, 104, 107, 112, 131,181,182 time-independent 16 Schrödinger-like equation 114 Schrödinger's cat state 205 Schrödinger's wave mechanics 11 Semiclassical quantization of chaos 15,200 resonance 93, 95

Index Sensitivity to initial conditions 1, 7, 181 Separatrix 4,164 bifurcation of 163 splitting 162, 163, 165,167,170,172 Shubnikov-de Haas oscillations 98 Slow degrees of freedom 117,122 Smale's horse-shoe mechanism 2, 6, 8, 10, 163, 180 S matrix 22, 25, 31, 35, 39, 55-57, 63, 83-89, 199, 200, 201 Soft wall model 98 Soliton 136 concentration of 139 condensation of 142 gas of 136,142 turbulence 136 Solitonic structures 136 Spectral rigidty 139 Spin-coherent state 192 Stability eigenvalues 5, 87 Stationary phase approximation 17, 109 Stochastic differential equation 128 Stokes constant 173,175, 176, 177 line 168,172, 175, 179 phenomenon 163,168,170,172,179 Stretching mechanism 8 Symbolic codings of periodic orbits 87, 202 dynamics 97,200 Symmetry induced factorization 89 Symplectic map 163,164, 180, 194 nonlinear 185 Symplecticity condition 164 Tangent map 67 Tight binding model 70 Time-continuous mesurement 204 Time-difference 162, 183 Time-discrete classical dynamics 162 non-unitary quantum dynamics 186 quantum dynamics 187 sympledic but still non-unitary 197, 204 unitary quantum dynamics 183 Time discretization 162- 164 of Heisenberg equation of motion 204

213 Time quantization 194 Topological pressure functions 81 Tori collapse of 160 invariant 17, 19 nonresonant 4 resonant 4 stability of 3 Trace formula 19-22, 29, 57, 80, 87, 98, 206 Transfer matrix 34 Translational lattice symmetry 35 Transmission amplitudes 27 coefficient 35, 57 Tunnelings over-barrier 33 resonant 31-32, 39 Zener 35, 38,108 Two-level cluster function 152,153 correlation function 152 Umbilic points of curved surfaces 107 Unitary quantum dynamics 183 Unitary transformation 25 Universal conductance fluctuations (UCF) 48, 59, 66, 71 Vandermonde determinant 145 Wave diffraction 29 Wavepacket 41 dynamics 10 Weak-localization 49 Whittaker function 56 Wiener process 128 Wien-Planck scaling formula 19 Wigner distribution 139,147 level-spacing 136 representation 10 semicircle law 146 Winding Demkov model 113 Windingnumber 4, 18, 24

Fundamental Theories of Physics 67. Yu. L. Klimontovich Statistical Theory of Open Systems. Volume 1: A Unified Approach to Kinetic Description of Processes in Active Systems. 1995 ISBN 0-7923-3 199-0; Pb: ISBN 0-7923-3242-3 68. M. Evans and J.-P. Vigier: The Enigmatic Photon. Volume 2: Non-Abelian Electrodynamics. 1995 ISBN 0-7923-3288-1 ISBN 0-7923-3340-3 69. G. Esposito: Complex General Relativity. 1995 70. J. Skilling and S. Sibisi (eds.): Maximum Entropy and Bayesian Methods. Proceedings of the Fourteenth International Workshop on Maximum Entropy and Bayesian Methods. 1996 ISBN 0-7923-3452-3 71. C. Garola and A. Rossi (eds.): The Foundations of Quantum Mechanics - Historical ISBN 0-7923-3480-9 Analysis and Open Questions. 1995 72. A. Peres: Quantum Theory: Concepts and Methods. 1995 (see for hardback edition, Vol. 57) ISBN Pb 0-7923-3632-1 73. M. Ferrero and A. van der Merwe (eds.): Fundamental Problems in Quantum Physics. 1995 ISBN 0-7923-3670-4 ISBN 0-7923-3794-8 74. F.E. Schroeck, Jr.: Quantum Mechanics on Phase Space. 1996 75. L. de la Pena and A.M. Cetto: The Quantum Dice. An Introduction to Stochastic Electrodynamics. 1996 ISBN 0-7923-3818-9 76. P.L. Antonelli and R. Muon (eds.): Lagrange and Finsler Geometry. Applications to Physics and Biology. 1996 ISBN 0-7923-3873-1 77. M.W. Evans, J.-P. Vigier, S. Roy and S. Jeffers: The Enigmatic Photon. Volume 3: ISBN 0-7923-4044-2 Theory and Practice of the B(3) Field. 1996 78. W.G.V. Rosser: Interpretation of Classical Electromagnetism. 1996 ISBN 0-7923-41 87-2 79. K.M. Hanson and R.N. Silver (eds.): Maximum Entropy and Bayesian Methods. 1996 ISBN 0-7923-43 11-5 80. S. Jeffers, S. Roy, J.-P. Vigier and G. Hunter (eds.): The Present Status of the Quantum Theory of Light. Proceedings of a Symposium in Honour of Jean-Pierre Vigier. 1997 ISBN 0-7923-4337-9 81. Still to be published 82. R. Muon: The Geometry of Higher-Order Lagrange Spaces. Applications to Mechanics and Physics. 1997 ISBN 0-7923-4393-X 83. T. Hakioglu and A.S. Shumovsky (eds.): Quantum Optics and the Spectroscopy of Solids. Concepts and Advances. 1997 ISBN 0-7923-4414-6 84. A. Sitenko and V. Tartakovskii: Theory of Nucleus. Nuclear Structure and Nuclear Interaction. 1997 ISBM 0-7923-4423-5 85. G. Esposito, A.Yu. Kamenshchik and G. Pollifrone: Euclidean Quantum Gravity on ISBN 0-7923-4472-3 Manifolds with Boundary. 1997 86. R.S. Ingarden, A. Kossakowski and M. Ohya: Information Dynamics and Open ISBN 0-7923-4473-1 Systems. Classical and Quantum Approach. 1997 87. K. Nakamura: Quantum versus Chaos. Questions Emerging from Mesoscopic Cosmos. 1997 ISBN 0-7923-4557-6

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